ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS 40
MATROID
APPLICATIONS Edited by NEIL WHITE
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ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS 40
MATROID
APPLICATIONS Edited by NEIL WHITE
This volume, the third in a sequence that began with The Theory of Matroids and Combinatorial Geometries, concentrates on the applications of matroid theory to a variety of topics from engineering (rigidity and scene analysis), combinatorics (graphs, lattices, codes and designs), topology and operations research (the greedy algorithm). As with its predecessors, the contributors to this volume have written
their articles to form a cohesive account so that the result is a volume that will be a valuable reference for research workers.
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS EDITED BY G.-C. ROTA Editorial Board R.S. Doran, J. Goldman, T.-Y. Lam, E. Lutwak
Volume 40
Matroid Applications
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS 1
2 3
4 5
6
L. A. Santalo Integral Geometry and Geometric Probability G. E. Andrews The Theory of Partitions R. J. McEliece The Theory of Information and Coding: A Mathematical Framework for Communication W. Miller, Jr Symmetry and Separation of Variables D. Ruelle Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics
H. Minc Permanents
F. S. Roberts Measurement Theory with Applications to Decisionmaking, Utility, and the Social Sciences 8 L. C. Biedenharn and J. D. Louck Angular Momentum in Quantum Physics: Theory and Application 9 L. C. Biedenharn and J. D. Louck The Racah-Wigner Algebra in Quantum 7
Theory 10
J. D. Dollard and C. N. Friedman Product Integration with Application to Differential Equations
11
W. B. Jones and W. J. Thron
Continued Fractions: Analytic Theory and
Applications 12 13
N. F. G. Martin and J. W. England Mathematical Theory of Entropy G. A. Baker, Jr and P. Graves-Morris Pade Approximants, Part I: Basic Theory
14
G. A. Baker, Jr and P. Graves-Morris Pade Approximants, Part II:
Extensions and Applications E. C. Beltrametti and G. Cassinelli The Logic of Quantum Mechanics G. D. James and A. Kerber The Representation Theory of the Symmetric Group 17 M. Lothaire Combinatorics on Words 18 H. O. Fattorini The Cauchy Problem 19 G. G. Lorentz, K. Jetter and S. D. Riemenschneider Birkhoff Interpolation 20 R. Lidl and H. Niederreiter Finite Fields 21 W. T. Tutte Graph Theory 22 J. R. Bastida Field Extensions and Galois Theory 23 J. R. Cannon The One-Dimensional Heat Equation 24 S. Wagon The Banach-Tarski Paradox 25 A. Salomaa Computation and Automata 26 N. White (ed.) Theory of Matroids 27 N. H. Bingham, C. M. Goldie and J. L. Teugels Regular Variation 28 P. P. Petrushev and V. A. Popov Rational Approximation of Real Functions 29 N. White (ed.) Combinatorial Geometries 30 M. Pohst and H. Zassenhaus Algorithmic Algebraic Number Theory 31 J. Aczel and J. Dhombres Functional Equations in Several Variables 32 M. Kuczma, B. Choczewski and R. Get Iterative Functional Equations 33 R. V. Ambartzumian Factorization Calculus and Geometric Probability 34 G. Gripenberg, S.-O. Londen and O. Staffans Volterra Integral and Functional Equations 35 G. Gasper and M. Rahman Basic Hypergeometric Series 36 E. Torgersen Comparison of Statistical Experiments 37 A. Neumaier Interval Methods for Systems of Equations 38 N. Korneichuk Exact Constants in Approximation Theory 39 S. Sakai Operator Algebras in Dynamical Systems 15 16
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
Matroid Applications Edited by
NEIL WHITE University of Florida
AMBRIDGE
UNIVERSITY PRESS
CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo
Cambridge University Press The Edinburgh Building, Cambridge C132 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521381659
© Cambridge University Press 1992
This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1992
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication data p.
Matroid applications / edited by Neil White. cm. - (Encyclopedia of mathematics and its applications ; v. 40) Includes bibliographical references and index. ISBN 0-521-38165-7
1. Matroids. I. White, Neil. II. Series. QA 166.6.M38 1992 511'.6-dc2O 91-27184 CIP ISBN 978-0-521-38165-9 hardback
Transferred to digital printing 2007
CONTENTS
List of Contributors Preface 1
Matroids and Rigid Structures Walter Whiteley
1.1 Bar Frameworks on the Line - The Graphic Matroid 1.2 Bar Frameworks in the Plane 1.3 Plane Stresses and Projected Polyhedra 1.4 Bar Frameworks in 3-space 1.5 A Matroid from Splines 1.6 Scene Analysis of Polyhedral Pictures 1.7 Special Positions 1.8 Conclusions 1.9 Historical Notes Exercises References
2 Perfect Matroid Designs M. Deza 2.1 Introduction 2.2 Some Lattice Properties and Parameters of PMDs 2.3 Intersection Properties of the Hyperplane Family 2.4 PMDs of Rank 4 2.5 Triffids 2.6 Some Close Relatives of PMDs and Further Directions References
3 Infinite Matroids James Oxley 3.1 Pre-independence Spaces and Independence Spaces 3.2 B-matroids Exercises References
Contents
viii
4 Matroidal Families of Graphs J. M. S. Simoes-Pereira 4.1 Introduction 4.2 Matroidal Families of Graphs: a Definition 4.3 Countably Many Matroidal Families 4.4 Uncountably Many Matroidal Families 4.5 Digraphs and Infinite Graphs Exercises References
5 Algebraic Aspects of Partition Lattices Ivan Rival and Miriam Stanford 5.1 Introduction 5.2 The Lattice of All Partitions 5.3 Lattice Embeddings 5.4 Postscript Exercises References
6 The Tutte Polynomial and Its Applications Thomas Brylawski and James Oxley
6.1 Introduction 6.2 The Tutte Polynomial 6.3 T-G Invariants in Graphs 6.4 The Critical Problem 6.5 Linear Codes 6.6 Other Tutte Invariants Exercises References
7 Homology and Shellability of Matroids and Geometric Lattices Anders Bjorner
7.1 Introduction 7.2 Shellable Complexes 7.3 Matroid Complexes 7.4 Broken Circuit Complexes 7.5 Application to Matroid Inequalities 7.6 Order Complexes of Geometric Lattices 7.7 Homology of Shellable Complexes 7.8 Homology of Matroids 7.9 Homology of Geometric Lattices 7.10 The Orlik-Solomon Algebra 7.11 Notes and Comments Exercises References
8 Introduction to Greedoids Anders Bjorner and Giinter M. Ziegler 8.1 Introduction 8.2 Definitions and Basic Facts 8.3 Examples
91
91
92 96 98 101 102 104 106
106 107 112 118 119
120
123
123
124 136 162 181 188 197
215
226
226 228 232 238 242 248 252 255 259 264 267 270 281
284 284 286 293
Contents
ix
8.4 Structural Properties 8.5 Optimization on Greenoids 8.6 The Greedoid Polynomial 8.7 Antimatroids
302
8.8 Poset of Flats
327
8.9 Further Topics 8.10 Notes and Comments
308 313
319
335 343
Exercises References
348
Index
358
354
CONTRIBUTORS
Anders Bjorner Department of Mathematics Royal Institute of Technology S-100 44 Stockholm SWEDEN Thomas Brylawski Department of Mathematics University of North Carolina Chapel Hill, NC 27514 USA
M. Deza Tokyo Institute of Technology Department of Information Science Ookayama, Meguro-ku, Tokyo 152 JAPAN James Oxley Department of Mathematics Louisiana State University Baton Rouge, LA 70803 USA
Ivan Rival Department of Computer Science 34 Somerset Street E. University of Ottawa Ottawa, KiN 9B4 CANADA
J. M. S. Simoes-Pereira Departamento de Matematica Faculdade de Ciencias e Tecnologia Universidade de Coimbra Apartado 3008 3000 Coimbra, PORTUGAL
Miriam Stanford Department of Mathematics Queens University Kingston, Ontario K7L 3N6 CANADA Prof. Walter Whiteley 461 Putney St. Lambert, Quebec J4P 3B8 CANADA
Prof. Gunter M. Ziegler Department of Mathematics Augsburg University Universitatsstr. 2 D-8900 Augsburg
F.R. of GERMANY
PREFACE
This is the third volume of a series that began with Theory of Matroids and continued with Combinatorial Geometries. These three volumes are the
culmination of more than a decade of effort on the part of the many contributors, potential contributors, referees, the publisher, and numerous other interested parties, to all of whom I am deeply grateful. To all those who waited, please accept my apologies. I trust that this volume will be found to have been worth the wait. This volume begins with Walter Whiteley's chapter on the applications of
matroid theory to the rigidity of frameworks: matroid constructions prove to be rather useful and matroid terminology provides a helpful language for the basic results of this theory. Next we have Deza's chapter on the beautiful applications of matroid theory to a special aspect of combinatorial designs, namely perfect matroid designs. In Chapter 3, Oxley considers ways of generalizing the matroid axioms to infinite ground sets, and Simoes-Pereira's chapter on matroidal families of graphs discusses other ways of defining a matroid on the edge set of a graph than the usual graphic matroid method. Next, Rival and Stanford consider two questions on partition lattices. These lattices are a special case of geometric lattices and the inclusion of this chapter will provide a lattice-theoretic perspective which has been lacking in much current matroid research (but which seems alive and well in oriented matroids). Then we have the comprehensive survey by Brylawski and Oxley of the Tutte
polynomial and Tutte-Grothendieck invariants. These express the deletion-contraction decomposition that is so important within matroid theory and some of its important applications, namely graph theory and coding theory. Bjorner describes the homology and shellability properties of several simplicial complexes associated with a matroid; the complexes of independent sets, of broken circuits, and of chains in the geometric lattice. This chapter
and the previous one constitute a study of the deepest known matroid
xii
Preface
invariants. We conclude with an exposition by Bjorner and Ziegler of greedoids, a generalization of matroids that embody the greedy algorithm and hence are very useful in operations research. University of Florida
Neil L. White
1
Matroids and Rigid Structures WALTER WHITELEY
Many engineering problems lead to a system of linear equations - a represented matroid - whose rank controls critical qualitative features of the example (Sugihara, 1984; 1985; White & Whiteley, 1983). We will outline a selection of such matroids, drawn from recent work on the rigidity of spatial structures, reconstruction of polyhedral pictures, and related geometric problems.
For these situations, the combinatorial pattern of the example determines a sparse matrix pattern that has both a generic rank, for general `independent' values of the non-zero entries, and a geometric rank, for special values for the coordinates of the points, lines, and planes of the corresponding geometric model. Increasingly, the generic rank of these examples has been studied by matroid theoretic techniques. These geometric models provide nice illustrations
and applications of techniques such as matroid union, truncation, and semimodular functions. The basic unsolved problems in these examples highlight certain unsolved problems in matroid theory. Their study should also lead to new results in matroid theory.
1.1. Bar Frameworks on the Line - the Graphic Matroid We begin with the simplest example, which will introduce the vocabulary and the basic pattern. We place a series of distinct points on a line, and specify certain bars - pairs of joints which are to maintain their distance defining a bar framework on the line. We ask whether the entire framework is `rigid' - i.e. does any motion of the joints along the line, preserving these distances, give all joints the same velocity, acceleration, etc.? Clearly a framework has an underlying graph G = (V, E), with a vertex vi for each joint pi and an undirected edge {i, j} for each bar {p;, pj}. In fact, we describe the framework as G(p), where G is a graph without multiple edges or loops, and p is an assignment of points pi to the vertices v;. If this graph is not connected,
2
Walter Whiteley
Figure 1.1.
(a) 0 --0 (b) o
o-
o
0
0
0
then each component can move separately in the framework, and the framework is not rigid (Figure la). Conversely, a connected graph always leads to a rigid framework (Figure 1.1b), since each bar ensures that its two joints have the same motion on the line. This gives an informal proof of the following result. 1.1.1. Proposition. A bar framework G(p) on the line is rigid if and only if the underlying graph G is connected.
To extract a matrix, we make this argument a little more formal. Assume the joints p, move along smooth paths pi(t). The length of a bar IIp;(t)-p;(t)II and its square, remain constant. If we differentiate, this condition becomes
dt [pi(t) - pi(t)]' = [pi(t) - pi(t)] Cpi(t) - pi(t)] = 0. At t = 0, this is written (p i - pj)(p; - p;) = 0. If we have distinct joints on the line, so that (pi - pj) 0, this simplifies to (p; - pj) = 0. With this in mind, we define an infinitesimal motion of a bar framework on the line G(p) as an assignment of a velocity ui along the line to each joint
pi such that ui - ui = 0 for each bar {vi, vj }. For example, consider the framework in Figure 1.1c. The four bars lead to four equations in the unknowns u = (ut, u2, u3, u4, u5): 1
0 0 0
0 1
0 0
-1 -1 1
1
0 0
-1 0
0 0 O
-1
rvl
u2 u
3
4 U4
0 0 0 0
US
In general, this system of linear equations is written R(G, p) x u= 0, where the rigidity matrix R(G, p) has a row for each edge of the graph and a column
Matroids and Rigid Structures
3
for each vertex, and u` is the transpose of the vector of velocities. We note that R(G, p) is the transpose of the usual matrix representation for the graph
over the reals: the rows are independent in R(G, p) if and only if the corresponding edges are a forest (an independent set of edges in the cycle matroid of the graph). A trivial infinitesimal motion is the derivative of a rigid motion of the line - i.e. a translation with all velocities equal. These form a one-dimensional subspace of the solutions. An infinitesimally rigid framework on the line has only these trivial infinitesimal motions, so the rigidity matrix has rank I VI - 1.
This rank corresponds to a spanning tree on the vertices, or a basis for the cycle matroid of the complete graph on I VI vertices. This proves the following
infinitesimal version of Proposition I.I.I.
1.1.2. Proposition. A bar framework G(p) on the line is infinitesmially rigid if and only if the underlying graph G is connected.
1.2. Bar Frameworks in the Plane A bar framework in the plane is a graph G = (V, E) and an assignment p of points pi e ll 2 to the vertices vi such that pi 0 pp if {i, j } e E. If we differentiate
the condition that bars have constant length in any smooth motion, we have
dt [pi(t) - p,(t)]2 = [pi(t) - p,(t)]. [pl(t) - p;(t)] = 0. Accordingly, an infinitesimal motion of plane bar framework is an assignment u of velocities ui a J2 to the joint such that
(pi - p) (ui-uj)=0 for each {i, j}eE. A plane bar framework is infinitesimally rigid if all infinitesimal motions are trivial: ui = s + f(pi)1, where s is a fixed translation vector, (x, y)1= (y, - x)
rotates the vector 90° counterclockwise, and fl(pi)' represents a rotation about the origin. (These infinitesimal rotations and translations are the derivatives of smooth rigid motions of the plane.)
The system of equations for an infinitesimal motion has the form R(G, p) x u= 0, where the rigidity matrix R(G, p) now has a row for each edge of the graph and two columns for each vertex. The row for edge {i, j} has the form [0 0
...
0 0
pi - pj 0 0 ... 0 0 pj- pi 0 0 ... 0 0]
Walter Whiteley
4
Figure 1.2.
P4
Pi
P2
P3
"
(a)
(c)
(b)
1.2.1. Example. Consider the frameworks in Figure 1.2. The framework of Figure 1.2a gives the rigidity matrix x,-x3
0
0
x3-x1
Y3 _Y1
{1, 4) 2(x, -x2) 2(Yi -Y2)
0
0
0
0
{1, 5)
X,
YJ
0
0
0
0
0
0
-x,
{2, 3}
0
0
x2-x3
Y2 _Y3
x3-x2
x3 - Y2
0
0
0
0
{2, 4}
0
0
2(x2 - x1)
0
0
0
0
{2, 5}
0
0
x2
y2
0
0
0
0
{3, 5}
0
0
0
0
x3
y3
0
0
(1,3)
Y1 -Y3
+(Y2 - Yi)
0
0
2(x2 - X0 z(Y2 - YO
2(x - x2) 2(yi - y2)
0
0
0
0 -y1
-x2 -Y2 -x3 -Y3J
The rows of this matrix are dependent and have rank 6. This leaves a (10 - 6 = 4)-dimensional space of infinitesimal motions, including the non-trivial
motion shown in Figure 1.2b, which assigns zero velocity to all joints but p4, and gives p4 a velocity perpendicular to the bars at p4. Thus the framework is not infinitesimally rigid. The infinitesimal motion is not the derivative of some smooth path for the vertices. The framework is rigid - all smooth paths, or even continuous paths,
give frameworks congruent to the original framework. Figure 1.2c gives a similar framework which has the same infinitesimal motions, but is not rigid. These examples show that there is a difference in the plane between rigid frameworks and infinitesimally rigid frameworks. A non-rigid plane framework will have an analytic path of positions p(t) = (..., pi(t),...), with all bar lengths of p(t) the same as bars in p(0), but p(t) not congruent to p(0), for all 0 < t < 1
(Figure 1.2c). The first non-zero derivative of this path will be a non-trivial infinitesemal motion. However, the converse is false: many infinitesimal motions are not the derivative of an analytic path (recall Figure 1.2b). For any framework, the independence of the rows of the rigidity matrix induces
a matroid on the edges of the graph. If `rigidity' in a particular plane framework were used to define an independence structure on the edges of a
Matroids and Rigid Structures
5
graph, this need not be a matroid (see Exercise 1.6). Therefore, we will restrict
ourselves, throughout this chapter, to the simpler concepts of infinitesimal motions and infinitesimal rigidity. The space of trivial plane infinitesimal motions has dimension 3, for frameworks with at least two distinct joints. This space can be generated by two translations in distinct directions and a rotation about any fixed point. Thus an infinitesimally rigid framework with more than two joints will have an JEl by 21 VI rigidity matrix of rank 21 VI - 3. Our basic problem is to determine which graphs G allow this matrix to have rank 21 VI - 3 for at least some plane frameworks G(p). The independence structure of the rows of the rigidity matrix defines a matroid on the edges of the complete graph on the vertices. This matroid depends on the positions of the joints. If we vary the positions there are `generic' positions that give a maximal collection of independent sets (for example, positions where the coordinates are algebraically independent real numbers). At these positions we have the generic rigidity matroid for IVI vertices in the plane.
1.2.2. Example. Consider the framework in Figure 1.3a. With vertices as indicated we have the rigidity matrix (al, bl) 1 0 0 0 0 0 -1 0 0 0 0 0 (al, b2) 0 1 0 0 0 0 0 0 0 -1 0 0 1
0
0
0
0
0 0
0
0
-1
-1
(a2, bl) 0 0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
2
-1
0
0
0
0
0
0
0
1
1
-2 -1
2
0
0
(al, b3)
1
(a2, b2) 0
0
-1 -2
(a2, b3)1 0
0
-1
1
0
(a2, bl)1 0
0
0
0
1
(a3, b2)) 0
0
0
0
0
1
0
0
0
-1
0
(a3, b3) 0
0
0
0
1
-1
0
0
0
0
-1
0
0 0 1
The graph of the framework has IEI = 21 VI - 3, so the framework is infinitesimally rigid if and only if the rows are independent. This independence
can be checked by deleting the final three columns and seeing that the determinant of the 9 x 9 submatrix is non-zero. This framework is infinitesimally
rigid and the graph is generically rigid, and generically independent. Consider any realization with distinct joints a1, a2, a3, bl, b2, b3 on a unit
circle centred at the origin (Figure 1.3b). This has a non-trivial 'in-out' infinitesimal motion (Figure 1.3c):
for joints ai take the velocity ai = ai; for joints b1 take the velocity b;= -bj.
6
Walter Whiteley
Figure 1.3.
(C)
These velocities preserve the length of all bars (ai, ba), since
This infinitesimal motion is non-trivial. Letting 00 0 be the angle between the unit vectors at and a2, we show that the distance (al -a2) is changing instantaneously:
(at -a2)-(ai
1 + 1 -2cos0> 0.
Thus this special position is not generic (see Exercise 1.9). We want to characterize the graphs of isostatic plane frameworks - minimal
infinitesimally rigid frameworks in the sense that removing any one bar introduces a non-trivial infinitesimal motion. These graphs, of size IEI = 21 VI - 3,
are the bases of the generic rigidity matroid `of the complete graph' on the set of vertices.
Thus an isostatic framework corresponds to a row basis for the rigidity matrix of any infinitesimally rigid framework extending the framework. The independence of such a set of edges is determined by maximal minors of the rigidity matrix. This independence is generic in the sense that these minors are non-zero polynomials in the positions pi. If such a polynomial is non-zero for some position G(p), then almost all q e ll "vlgive isostatic frameworks G(q) (see Section 1.7).
More surprisingly, for points where this matrix and all its minors have the maximal rank achieved for q e R21v", infinitesimal rigidity and any reasonable form of local rigidity actually coincide (see, for example, Exercise 1.7).
We note that throughout this chapter the generic matroids defined on the complete graph of I VI vertices are symmetric on the vertices - any permutation
of the vertices does not change the independence of a set of edges. As a convention, we write the attached vertices for a subset of edges E' as V.
Matroids and Rigid Structures
7
1.2.3. Theorem. For a graph G, with at least two vertices, the following are equivalent conditions: (i) G has some positions G(p) as an isostatic plane framework; (ii) IEl = 21 VI - 3 and for all proper subsets of edges E' incident with vertices
V', IE'I<21V'I-3; (iii) adding any edge to E (including doubling an edge) gives an edge set covered by two edge-disjoint spanning trees.
(ii): For an isostatic plane framework G(p) on at least two vertices, the rows of the rigidity matrix have rank IEl = 21 VI - 3. If any proper subset of edges has IE'I > 21 V'I - 3, the corresponding rows are dependent. Since G(p) is independent, we conclude that IE'I < 21 V'I - 3 for all proper subsets. (ii)e>(iii): The count f (E') = 21 V'I - 3 defines a non-decreasing semimodular Proof. (i)
function on sets of edges, which is non-negative on non-empty sets (see Exercise 1.1). This semimodular function defines a matroid by the standard property: E is independent if and only if IE'I < f(E') for all proper subsets E'. (1.1) This count has the form f (E') _ (21 V'J - 2) - 1 which shows that the matroid for f is a Dilworth truncation of the matroid defined by the semimodular function g(E') = 2(1 V'I - 1). In turn, the semimodular function g represents a matroid union of two copies of the matroid given by the semimodular function h(E') = I VI - 1 (the cycle matroid of the graph). Thus a graph is independent in the matroid of f if and only if adding any edge (including doubling an edge) gives a graph covered by two edge-disjoint forests. Before we prove (iii)=>(i), we need a lemma about a simpler matrix that has rank 21 VI - 2 (matching the function g). For a graph G = (V, E), including possible multiple edges, a 2 -frame G(d) is an assignment of directions de E t2 to the edges. An infinitesimal motion of the 2 -frame G(d) is an assignment of velocities u; E l 2 to the vertices such that de (u, - u;) = 0 for every edge e joining vi and v j (i < j). This system of equations defines the rigidity matrix R(G, d) for the 2-frame. 1.2.4. Lemma. The rows of the rigidity matrix of a generic 2 frame G(d) are independent if and only if G is the union of two edge-disjoint forests.
Proof. Take the two forests Ft and F2. For all edges in the first forest, we assign the direction (1, 0). For all edges in the second forest, we assign the direction (0, 1). If we reorder the rows and columns of this rigidity matrix, placing all second columns of vertices to the right, and all rows for the second
forest at the bottom, we have a pattern: C[FI] [0]
[0] 1
[F2]
8
Walter Whiteley
where [F1] and [F2] are the standard matrices representing the two forests over the reals. Thus the blocks [F1] and [F2] have non-zero minors on all their rows, and the entire matrix also has a non-zero minor on all the rows. We conclude that the rows are independent. Conversely, if the rows are independent, we can reorder the columns as above. The independence guarantees a non-zero minor on all the rows. Using a Laplace expansion on the two blocks, there are non-zero minors on complementary sets of rows. Each of these non-zero minors on the matrix representing the graphic matroid must correspond to a forest, as required. Proof of Theorem 1.2.3 (continued). (iii) => (i) Assume that adding any edge to E (including doubling an edge) gives an edge set covered by two edge-disjoint
forests. We show that this gives independent rows in the rigidity matrix for some (almost all) choices of the points. Take a 2-frame G(d) with algebraically independent directions for the edges. By our assumption, adding any edge (or doubling any edge) between any pair of vertices gives an independent 2-frame E*. Therefore, the 2-frame on E has an infinitesimal motion u1, that has different velocities on the two vertices of the added edge. Taking linear combinations of these uii there is an infinitisemal motion u that assigns distinct velocities ui = (si, ti) to each of the vertices of G. To create the independent framework G(p), we set pi = (- ti, si ). Since uj, we have pi pj for each edge { i, j }, as required in a framework. We ui claim that the rigidity matrix of G(p) has rows parallel to the rows of the original 2-frame G(d). Clearly t;,
ti-t;)=0=de-(ui-u;).
Since ui - uj 0, this means (pi - pj) = fede for some non-zero scalar fle. The rigidity matrix of the framework is equivalent to the rigidity matrix of the independent 2-frame. We conclude that a set of edges satisfying condition (1.1) has been realized as an independent (therefore isostatic) bar framework. (The infinitesimal motion u of the 2-frame is a rotation of this framework around the origin.) This completes the proof. Figure 1.4 shows some other examples of the graphs of isostatic frameworks in the plane (Figure 1.4a) and graphs of circuits in the plane generic rigidity matroid (Figure 1.4b) with IEI = 21 VI - 2, and IE'I < 21 V'I - 3 for proper subsets.
The semimodular count of Theorem 1.2.3 (ii) converts to a criterion for graphs of infinitesimally rigid frameworks. We state the theorem without proof.
1.2.5. Corollary. (Lovasz & Yemini, 1982) A graph has realizations as an infinitesimally rigid plane framework if and only if for every partition of the
Matroids and Rigid Structures
9
edges into non-empty subsets (... Es ...), with vertices V' incident with the edges E',
Y(21V'I-3),21 VI-3. Figure 1.5 gives a simple example of a 5-connected graph, in a vertex sense, which is never infinitesimally rigid by this criterion: take the eight K5 graphs as sets of the partition, and all other edges as singletons and apply Corollary
1.2.5. However, every graph which is 6-connected in a vertex sense is generically rigid (Exercise 1.16).
The dependence of rows in the rigidity matrix, or dependence of bars in the framework, also has a physical interpretation. A self-stress on a framework
is an assignment of scalars wii to the bars {pi, pj} such that for each joint pi there is an equilibrium (Figures 1.6a, b): Figure 1.4. (a) IEI = 21 V1- 3, IE'I < 21 V1- 3; (b) IEI = 21 VI - 2, IE'I < 21 V'I - 3. 11
11
11
11
11
11
0
(a)
Figure 1.5.
11
Walter Whiteley
10
Figure 1.6.
c012(P1-p2) 0)12(P2-P1)
^
0
Tension
w14(p4-p1) 014(P1-P4) 0 Compression
(c)
4
0 (f)
(sum over all edges { i, j } incident with pi).
coij(pj - pi) = 0
Thus a self-stress is equivalent to a row dependence. If coif < 0, we interpret
this as a compression in the bar - a force coj% - pi) at pi, and a force coij(pi - p3) at pj. If coin > 0, this is a tension in the bar (Figure 1.6c).
In the same spirit, the row space of the rigidity matrix is interpreted as the space of loads Li resolved by forces in the bars of the framework (Figure 1.6d, e, f):
Li +
wi;(p; - pi) = 0 (sum over all edges { i, j } incident with pi).
These resolved loads satisfy an additional property of global static equilibrium, defined below (see Exercise 1.17). Thus an equilibrium load is an assignment
L, of vectors to the vertices that satisfies the three equilibrium equations Y Li = O i
and Y Li x pi = 0 i
where x represents a cross product in 3-space. A framework is statically rigid if all equilibrium loads on its joints are resolved. We note that a single point is trivially both infinitesimally rigid and statically rigid in the plane. A single bar has only a one-dimensional space of equilibrium loads: a(p, - p2) at p, and 0(p2 - p,) at p2. Since these are
Matroids and Rigid Structures
11
resolved, the bar is statically rigid and infinitesimally rigid. Moreover, the equilibrium loads on any framework with at least two distinct joints form a space of dimension 21 VI - 3, and all resolved loads are equilibrium loads (Exercise 1.17). These observations prove the following. 1.2.6. Theorem. For a bar framework G(p) in the plane, on at least two joints, the following are equivalent conditions: (i) the framework is statically rigid; (ii) the rigidity matrix has rank 21 VI - 3; (iii) the framework is infinitesimally rigid.
Such dual static concepts arise for the infinitesimal mechanics of most types of structure, giving dual static and infinitesimal mechanical theories for the row and column ranks of the corresponding matrix. Of course the non-zero self-stresses correspond to the dependences of the underlying matroid on the edges.
1.3. Plane Stresses and Projected Polyhedra There is a classical theorem that interprets self-stresses in a framework with a planar graph as instructions for building a `spatial polyhedron' with this projection. The ideas of the proof are simple, but we need a few topological definitions for our `spherical polyhedra'. We begin with a simple class of graphs, for which we can associate the face, edge, and vertex structures of abstract polyhedra. A planar graph is a
graph G that can be drawn in the plane with the edges as straight lines, disjoint except at shared vertices. A graph is vertex 3-connected (or 3-connected
for short) if deleting any two vertices leaves the graph connected. A planar drawing of a 3-connected planar graph G = (V; E) divides up the plane into a set of polygonal discs, and a single infinite region with a polygonal hole. These polygonal regions oriented counterclockwise, and the infinite region, with its polygonal hole oriented counterclockwise, form the oriented faces, (..., f h, ...), of the associated polyhedron. (Note that a theorem of graph theory
says that, up to reversing all these orientations, the face structure for a 3-connected planar graph is unique.) We call this collection of vertices, edges, and faces a 3-connected abstract spherical polyhedron S = (V, E, F).
1.3.1. Example. Figures 1.7a and 1.7b show two planar drawings of the same 3-connected graph. The reader can check that they define the same oriented faces. Figure 1.7c shows a planar graph that is not 3-connected: removing vertices a and c, or b and c, separates the graph. The results of this section actually extend to these 2-connected planar graphs - see Exercise 1.18.
Walter Whiteley
12
Figure 1.7.
(a)
(b)
(c)
We are interested in projections of spatial polehydra. We assume that we are projecting vertically down into the xy-plane, and that no face is a vertical plane. Thus the face planes can be written Ax + By + C = z, or (A, B, C) for short.
A spatial polyhedron S(Q, q) is an assignment of a plane Q" = (A", B", C")
to each face f", and points qj _ (xi, yj, zj) to each vertex vj, such that for each vertex Vk on a face f'
A`xk+B`yk+C`=Zk For convenience, we also assume that for any edge {vj, vk } of the polyhedron, qj qk. We do not assume that the two faces at an edge have to have distinct
planes, nor that the faces have to be topological discs, convex, etc. in space. A spatial polyhedron S(Q, q) projects to a plane framework G(p) if G is the graph of edges and vertices of S, and if for each vertex with qj = (xi, yj, z), p; = (x1, Y,)
1.3.2. Example. What geometric condition(s) on the drawings in Figure 1.8
correspond to a projected polyhedron? If a spatial polyhedron has three distinct planes for the faces aa'b'b, bb'c'c, and cc'a'a, these planes intersect in a single point, even in the projection. Therefore the three edges aa', bb', and
cc' must have a common point. The reader can mentally reconstruct the triangular prism over Figure 1.8a. Figure 1.8b is a generic drawing, and the only spatial polyhedra are trivial, with all faces in the same plane. Figure 1.8c corresponds to a 'degenerate' spatial polyhedron, which assigns one plane to face abc, and another plane to all other faces. This object satisfies our definition and is non-trivial.
Consider a plane framework on the graph of the polyhedron G(p), with distinct vertices for each edge. A polyhedron in space that projects to this framework assigns a plane z = A"x + By + C" to each face f". If two faces f", f ` share an edge over the line through {p,, pk }, we have the conditions
Matroids and Rigid Structures
13
Figure 1.8.
(a)
Ahxj+B''yj+Ch=A`xj+B`yj+C` and
A""xk+B"Yk+Ch=Aixk+B`Yk+C` or equivalently, by Cramer's rule, (Ak, Bk, Ch) - (A`, B`, C`) = wjk((YJ - Yk), (xk - xi), (xjYk - xkYJ))
for some scalar wjk. We use the orientation of the polyhedron to associate an order of the faces (h, i) with the order of the vertices (j, k), following the order of vertices in the cycle of face h. This orientation also associates (i, h) with (k, j). This generates a consistent scalar wjk = a)kj for each edge. The two planes are distinct if and only if cojk 0 0. If we follow a cycle of faces and edges around a vertex vi of the spatial polyhedron, the first face equals the last face. This gives the equation Y-wjk((Yj - Yk), (xk - x j), (x jyk - xk y j)) = (0, 0, 0)
(sum over the cycle of edges).
These scalars look suspiciously like a self-stress on the framework G(p). 1.3.3. Maxwell's theorem. Given a spatial spherical polyhedron S(Q, q) projecting
to the plane framework G(p), there is a self-stress of the framework G(p) such that an edge separates faces in distinct planes in the polyhedron if and only if the self-stress has a non-zero scalar on the edge.
Proof. We have seen that a spatial polyhedron leads to scalars wjk satisfying Y_ wjk(yj - Yk, xk - xj, xjyk - xk yj) = (0, 0, 0) (sum over the edges at vj). Clearly this implies that Y_wjk(xk - xj, yk - yj) = (0, 0) (sum over the edges at vj). Therefore the projection of a polyhedron gives a self-stress that is non-zero on exactly the edges separating distinct planes.
Walter Whiteley
14
1.3.2. Example (continued). By counting vertices and edges, it is found that the graph of Figure 1.8 is generically independent - hence a generic framework (Figure 1.8b) is not the projection of a spatial polyhedron. If the three edges
aa', bb', and cc' have a common point, Figure 1.8a, the spatial triangular prism proves that the framework has a self-stress that is non-zero on all edges. Figure 1.8c has a self-stress, non-zero only on edges of the collinear triangle abc. This corresponds to the degenerate spatial polyhedron that assigns one
plane to face abc and another plane to all other faces.
We have a converse to Maxwell's theorem. The next lemma presents a crucial property of all self-stresses, which we need in the proof of this converse.
1.3.4. Cut lemma. Given a framework G(p) with a self-stress w, and a set of bars (a;, b.) such that they separate the graph, with all a; in one component, and all b; in other components, then D w1(b1 - a1) = 0
and Y w;(b; x a;) = 0.
Proof. We have used the shorthand (b1 x a,) for the third component of (b,, 0) x (a,, 0). For a cut set that isolates a single vertex vj, we have (0, 0, 0) x (x3, yj, 0) = Y_ wjk(xk - xj, Yk - yj, 0) x (xi, Yj, 0) k
= Y_ wjk(0, 0, xiYk - xkYj) k
as required. Therefore:
Y,wjk(yj-Yk, xk - xj, xjyk-xkyj)=(0, 0, 0) (sum over the edges at vi). k
Take the component created by the cut set. If we add these equations over all vertices of the component, the terms cancel on all edges joining two ends in the component containing all the a1, and leave the required sums on the cut set (a;, bi).
1.3.5. Converse of Maxwell's theorem. Given a self-stress on the plane framework G(p) built on the graph of a spherical polyhedron, there is a spatial polyhedron projecting to G(p) such that an edge separates faces in distinct planes in the polyhedron if and only if the corresponding self-stress is non-zero on the edge.
Proof Assume that we have a self-stress (..., (ojk, ...) on the framework G(p). We choose the arbitrary plane z = A°x + B°y + C° for an initial face f °. For every other face f" we take a face-edge path II from f ° to f" and define the
plane for f" by
Matroids and Rigid Structures
15
(An, B", C")=(A°, B°, C°)+>wjk(Yi-Yk, xk - xj, xjYk - xkYj) n
We must prove that this answer is well defined, that is, independent of the path used. Consider two different paths n and H' from f ° to f". Together, the path II, -II' forms a face-edge cycle on the polyhedron (i.e. a cycle on the dual graph). Such a cycle is the disjoint union of simple cycles, and each simple cycle on a spherical polyhedron is a cut set satisfying Lemma 1.3.4. Summing over all these oriented cycles simultaneously: [s wjk(xj - xk, Yj - Yk, xjYk - xkYj)
H
- Y_ wjk(Xi - xk. Yj - Yk, xjYk - xkY,) = (0, 0, 0). n,
It is a simple exercise to rewrite the order and the signs of coordinates in this sum, giving: >wjk(Yj-Yk, xk-xj, xjYk-xkYj)= Y_ wjk(YJ-Yk, xk-xj, xjyk-xkYj). n,
H
This proves that the plane for face f" is well defined and that these spatial planes have the required incidences in space over the vertices to form a spatial polyhedron projecting to G(p).
For any edge -jk of the polyhedron, we still have (Ah, Bh, Ch) - (A`, B', C`) = wjk(Yj - Yk, Xk - xj, xjyk - xkYj).
Therefore the edge separates two distinct planes if and only if there is a non-zero scalar in the self-stress. 1.3.2. Example (completed). Consider the frameworks in Figure 1.8. We saw
that Figure 1.8a was the projection of a polyhedron with distinct faces hence it has a self-stress. Figure 1.8b is not the projection of a polyhedron - so it must be independent, by Theorem 1.3.5. The self-stresses on a framework G(p) actually form a vector space. The polyhedra that project to this framework, with non-vertical faces and with a face f ° held in a fixed plane z = 0, also form a vector space. (Note that we
include polyhedra with all faces coplanar, corresponding to the zero self-stress.) Theorems 1.3.3 and 1.3.5 actually give an isomorphism of these spaces.
1.3.6. Corollary. The space of spatial polyhedra over a plane drawing of the graph G(p), with the plane z = 0 assigned to a fixed face f °, is isomorphic to the vector space of self-stresses of G(p).
Walter Whiteley
16
1.4. Bar Frameworks in 3-space The definitions of a framework, the rigidity matrix, and infinitesimal rigidity in the plane easily generalize to bar frameworks in P3. Trivial infinitesimal motions now have the form u; = s + r x p.. These infinitesimal motions form a space of dimension 6 for any framework whose joints are not all on a line, a space generated by three non-coplanar translations and by three non-coplanar
rotations about any point. Thus a spatial framework with more than two joints is infinitesimally rigid if and only if the rigidity matrix has rank 31 VI - 6.
For statics in 3-space we have the similar modification that equilibrium loads satisfy the six equations ZiL1= 0 and Yi L; x p; = 0. These equations are independent provided that the joints are not all collinear. The row space of the rigidity matrix still corresponds to the resolved equilibrium loads. We also observe that a framework on the line with more than two joints cannot be statically or infinitesimally rigid even in the plane. These observations
lead directly to 1.4.1. Theorem. For bar framework G(p) in space, with at least three joints, the following are equivalent conditions: (i) G(p) is infinitesimally rigid; (ii) the rigidity matrix RG(p) has rank 31 VI = 6; (iii) G(p) is statically rigid. 1.4.2. Corollary. If a framework in space with at least three joints is isostatic then IEI = 31 VI - 6 and IE'I <, 31 V'l - 6 for all subsets of at least two edges.
When we seek the converse to this corollary, we run into a fundamental problem: some graphs that satisfy the count are never realized as isostatic
frameworks. Figure 1.9a gives a simple example, which clearly has a non-trivial motion based on rotations around the `hinge' ab. If we take two copies of K5, which are generic circuits in space (Figure 1.9b), and do a circuit exchange on the common bar ab we obtain this generic circuit. Tay (1986) surveys other generic circuits in 3-space that are not infinitesimally rigid (Figure 1.9c, Exercise 1.22). An alternative vision of our difficulty comes from the semimodular function f (E) = 31 VI - 6 on sets of edges. Since f (E) is negative on single edges, this function does not directly define a matroid. Instead we ask the more subtle question:
Is there a maximal matroid M(G) on graphs that has rank 31 VI - 6 on all complete graphs of at least three vertices? Maximal means that if E" is independent in some matroid on graphs satisfying
this count, then E" is independent in M.
Matroids and Rigid Structures
17
Figure 1.9.
(a)
(b) (c)
1.4.3. Graver's conjecture. The rigidity matroid in 3-space is maximal: any independent set in another matroid on finite graphs with rank 31 VI - 6 on each complete graph of I VI > 2 vertices will be independent in the rigidity matroid.
This question can be approached through a set of inductive constructions building up the number of vertices in an isostatic set. The graph of an isostatic
framework in 3-space that is larger than a triangle has a vertex of valence 3, 4, or 5. The edge set satisfies IEI = 31 VI - 6 and IE'I < 31 V'I - 6 for all subsets
of at least three vertices. Since each edge is incident with two vertices, we have 2IEI = Yi val(vi) < 61 VI. There must be vertices of valence less than 6. On the other hand, if we remove vertex vi attached to edges Ei, leaving at least three vertices V, we have
IE'I<3IV'I-6 and IE'uEiI=IE'I+IEiI=3IVI-6=3IV'I-6+3. Thus, if the graph is bigger than a triangle, each vertex must have valence of at least 3. Accordingly, the inductive techniques describe the addition and deletion of 3-, 4-, or 5-valent vertices. Unfortunately the techniques we outline remain incomplete for 5-valent vertices in 3-space. 1.4.4. Proposition. If an isostatic bar framework in 3-space has three non-collinear
joints P1, p2, p3, then inserting a 3-valent joint po out of this plane with bars (POI Pt), (POI P2), (p0, P3) creates an isostatic bar framework. Conversely, given any isostatic framework in 3-space with a 3-valent joint,
then deleting this joint with its bars leaves an isostatic bar framework (Figure 1.10a).
Walter Whiteley
18
Figure 1.10.
Proof. (1) For convenience, we assume that the added vertex vo is at the left side of the rigidity matrix, and the added bars are on the top:
Pi-Po PO-Pi P2 - PO
L
0
0
0
...
0
PO - P2
0
...
0
P3-Po
0
0
0
...
...
Po-P3 ... 0 ...
...
.. -1
It is clear that this vertex adds rank 3 to the lower rigidity matrix if and only if the three vectors pt - POI P2 - po, and P3 - po are independent, that is, the four points are not coplanar. (2) Conversely, removing a 3-valent joint keeps the independence of the rows of the rigidity matrix and keeps IE'I = JEl - 3 = 31 VI - 9 = 31 V'l - 6.
1.45. Proposition. If an isostatic bar framework in 3-space has four non-coplanar joints P1, P2, P31 P4 and a bar (pt, p4) then removing this bar and inserting a joint po with bars (po, pi), (POI P2), (POI P3), (POI P4) creates an isostatic bar framework for almost all positions of po. Conversely, given any isostatic bar framework in 3-space with a 4-valent joint po attached to four non-coplanar joints, then there is a bar (pi, pj) among the vertices adjacent to po such that deleting this joint and inserting (p;, pj) leaves an isostatic bar framework (Figure 1.10b).
Proof. (1) Again, we add the vertex vo at the left side of the rigidity matrix, with three bars to p1, p2, and p31 so that pt - po is parallel to Pt - P4:
Matroids and Rigid Structures
19
Pi - Po
Po - Pi
0
0
0
P2 - Po
0
Po - P2
0
0
P3 - Po
0
0
PO - P3
0
0
P4 - P1
0
0
Pt-P4
0
0
It is clear that this vertex adds rank 3 to the lower rigidity matrix if and only if the three vectors p, - po, p2 - po, and p3 - po are independent. However, we can now row reduce with row I and row IV to get the desired independent rigidity matrix: Pt - Po
Po - Pt
0
0
0
P2 - Po
0
PO - P2
0
0
P3 - PO
0
0
PO - P3
0
P4 - Po
0
0
0
PO - P4
0
0
Since we have IE'I = IEI + 4 -1= 31 VI - 3 = 31 V'I - 6, the framework is isostatic.
(2) Conversely, deleting a 4-valent joint po leaves
IE'I= =IEI-4=3IVI-10=3IV'I-7 independent bars. If some bar (pi, pd), 1 5 i, j < 4, is independent, the proof
is complete. If not, all six such bars are already implicit - as linear combinations of the remaining bars. These six implicit rows, plus the original four to po give a set of IE'I = 10 rows on five joints. This set with IE'I > 31 V'I - 6 rows is dependent - contradicting the assumption that the original framework was isostatic. Q
A 5-valent joint can be replaced by two bars in several different patterns (Figure 1.11a, Exercise 1.24). We have conjectured that the replacement procedures of Figure 1.1 lb preserve isostatic frameworks for generic positions of the joints. This would imply the basic Conjecture 1.4.3. In the absence of a general characterization, we present another inductive technique that covers some higher valent vertices in triangulated polyhedra
(Figure 1.12). Given a graph G with a vertex v, incident to the edges (1, 2), (1, 3), (1, 4), ..., (1, k), (1, k + 1), ..., (1, k + m), then a vertex split of v,
on the edges (1, 2), (1, 3) is the modified graph with edges (1, 4), ..., (1, k)
removed and an added vertex po incident with new edges
(0, 1),
(0, 2), (0, 3), (0, 4), ..., (0, k).
1.4.6. Vertex splitting theorem. Given an independent framework G(p) in 3-space with a joint p and all incident bars (1, 2), (1, 3), (1, 4), ..., (1, k),
Walter Whiteley
20
(1, k + 1), ..., (1, k + m) with p1 - p2 not parallel to P1 - p3, then for any k + m > 2, the new framework on the vertex split on (1, 2), (1, 3) is independent for almost all positions for the new joint po.
Proof. We choose, as a limiting initial case, to add po at p1, with the `bar' (Po' p1) assigned a direction d01 not in the plane of
d02=P2-PO=P2-Pl =d12 and d03 =P3-PO=P3-P1 =d13 This is not a bar framework, but a `3-frame' with a rigidity matrix - and it is the limit of frameworks with variable po (see Exercise 1.43). This creates the following matrix: d01
- do,
0
0
0
d02
0
-d02
0
0
d03
0
0
- d03
0
0
d12
-d12
0
0
0
d13
0
-d13
0
0
Pa - PI
0
0
P1 - Po
L0 Figure 1.11.
OR h BOTH
Figure 1.12.
Matroids and Rigid Structures
21
It is clear that this vertex adds rank 3 to the lower rigidity matrix if and only if the three vectors dol , d02 = p2 - po, and d03 = P3 - po are independent.
If we add row IV to row II, as well as row V to row III, we get the row equivalent matrix d01
-dot
0
0
0
d02
-d02
0
0
0
d03
-d03
0
0
0
0
d12
d12
0
0
0
d13
0
-d13
0
0
P4-p1
0
0
P1 - P4
0
...
With these first three independent rows, and with p0 - p4 = p1 - p4, we can row reduce on rows such as row VI, to change the non-zero entries to:
IN - Po 0 0 0 Po - P4
01
We can then return the top five rows to their original pattern, giving a matrix for the graph of the vertex split. Finally since this matrix is independent, it remains independent if we move p0 to any nearby position with p1 - po = d01 This completes the proof.
The converse is not true: if we split a vertex on two edges, we can turn a generically non-rigid graph (Figure 1.13a) into a generically rigid graph (Figure 1.13b). Figure 1.13c proves the generic rigidity of Figure 1.13b by a sequence of vertex splits from a tetrahedron which include one application of Proposition 1.4.5.
A selected vertex split on a triangulated surface will create a larger triangulated surface of the same topological type. Let v be a vertex of a triangulation of a 2-manifold (Figure 1.14a). The star of v is the union of the closed triangles meeting v, and the link of v is the simple polygon of edges in the star that do not contain v. The link can be split into two simple paths M and N, meeting at vertices x and y (Figure 1.14b). A vertex split of the triangulation at the vertex v consists of replacing the triangles of the star of v by two new vertices v' and v" and the triangles determined by v' and the edges of M, v" and the edges of N, and the two triangles v'v"x, v'v"y (Figure 1.14c).
This vertex split in a polyhedron can be reversed by shrinking an edge
pop, and identifying the vertices - provided that this edge is not part
22
Walter Whiteley
Figure 1.13.
Figure 1.14.
Matroids and Rigid Structures
23
of a non-facial triangle of the surface (Figure 1.14d) and the 2-manifold is not the tetrahedron. We illustrate this process of vertex splitting within the class of triangulated surfaces with a quick proof that all triangulated spherical polyhedra are generically rigid.
1.4.7. Generic Cauchy's theorem. Every triangulated spherical polyhedron is generically rigid in 3-space.
Proof. We prove this by induction on Vi, the number of vertices in the triangulated sphere. If I VI = 4, we have a tetrahedron, which is generically rigid in 3-space. Assume that generic rigidity holds for all triangulated spheres with 4 vertices. We first show
that there is a `shrinkable edge'. Any non-shrinkable edge e is part of a non-facial triangle separating a disc of N interior vertices from the remaining
triangulated sphere. We claim that there is a shrinkable edge inside each such disc. If N = 1, this vertex is 3-valent, and every interior edge is shrinkable.
Assume a shrinkable edge occurs for N = k. If a disc has k + 1 vertices, then there is an interior edge e. If e is shrinkable, the proof is complete. If not, e is part of a non-facial triangle - which separates the disc, with the previous boundary triangle on one side. The other component is a triangulated disc with fewer than k interior vertices, which must contain a shrinkable edge.
Take any shrinkable edge in the triangulated sphere with I VI vertices. Shrinking this edge creates a smaller generically rigid triangulated sphere. Splitting back along this edge, Theorem 1.4.6 shows that the initial triangulated sphere is also generically rigid in 3-space.
In the exercises we offer a few other examples that can be proven by similar inductions (Figure 1.15, Exercises 1.23, 1.27). However, the central problem of characterizing the generic rigidity matroid in 3-space remains unsolved. Figure 1.15.
(a)
(b)
Walter Whiteley
24
1.5. A Matroid from Splines Consider a planar drawing of a 3-connected planar graph, with a triangular boundary. In Section 1.3 we saw that this drawing is the projection of a
spatial polyhedron if and only if the drawing has a self-stress as a bar framework. If we ignore the exterior triangle (which is always a plane), such a spatial polyhedron is a globally continuous (C°) function that is piecewise linear over the plane regions (the `faces'). In approximation theory such a function is called a bivariate C°-spline. Maxwell's theorem and its converse
give a correspondence between C°-splines and self-stresses for planar dissections of a boundary triangle. Consider the analogous problem of finding globally C1 functions over the planar drawing, which are piecewise second degree polynomials - bivariate C2-splines. We shall extract a central matroid for these splines that has a deep analogy to the rigidity matroid for spatial frameworks. We outline how
the techniques of sections 1.3 and 1.4 extend to this new matroid to characterize the dimension of this space of CZ-splines, for generic planar drawings with a triangular boundary. A plane geometric graph G(p) is a graph G = (V; E), with no loops or multiple edges, and an assignment p: V-> R2, p(vi) = pi, giving distinct points to the two ends of each edge. For our final application to splines these graphs
will be chosen to be planar embeddings of planar graphs. However, in the general theory, including the construction of some special splines, the graph
need not even be planar. A generic geometric graph uses algebraically independent real numbers for the coordinates of the vertices. For each edge we have a line with equation
p,,x+qijy+rid=(yi-yi)x+(xi-xj)y+(xjyi-xiyj)=0. The double-line vector for a directed edge (i, j) is the vector
D U =(p,ii 2piigi,,q)=((y,-yi)2, 2(yj-yi)(xi-xi), (xi-xj)2) ii
for i <j,
and
D = -D j
for
i <j.
A conic-dependence of a geometric graph is an assignment of scalars owij to the directed edges, (0ji = Coif, such that at each vertex pi there is a balance: coijD; = 0 (sum over edges incident with pi). A geometric graph is conic-independent if the only conic-dependence is trivial (all coif = 0).
These dependences correspond to a matrix equation owRC(G, p) = 0, where RC(G, p) is the IEI x 31VI conic-rigidity matrix for a geometric graph. For edges (1, 2) and (i, j) the rows of RC(G, p) have the form:
25
Matroids and Rigid Structures
[D12
-D12 000
000
...
[000 000 000
...
D
2
000
... ...
-D, 2
000] 000]
...
...
1.5.1. Example. Consider a plane triangle with vertices p1 = (0, 0), p2 = (0, 1), p3 = (1, 0). The edges of this triangle have the double line vectors
D12=((Y2-Yt)2, 2(Y2-Y1)(x1 -x2), (x1 -x2)=(1, 0, 0) D12
2
3 = (0, 0, 1)
D23 = (1, 2, 1)
This gives the conic-rigidity matrix
-1
0
0
0
0
0
1
0
0
0
0
0
-1
0
1
2
1
1
0
0
0
0
0
0
.
-1 -2 -1
Clearly this matrix is independent with rank 3, so there is only the trivial conic-dependence. Accordingly, the solution set to the homogeneous system RC(G, p) x u` = 0
has dimension 6. The reader can check that this solution space is spanned by the six independent vectors: T1= (1,0,0, 1,0,0, 1,0,0) T2 = (0,1,0, 0,1,0, 0,1,0) T3 = (0,0,1, 0,0,1, 0,0,1) T4 = (0,0,0, 0,1,0, 2,0,0) T5 = (0,0,0, 0,0,2, 0,1,0) T6 = (0,0,0, 0,0,1, 1,0,0)
The reader can also check that a collinear triangle gives a conic-rigidity matrix of rank 2: all the D, are parallel vectors, and there is a conic-dependence.
1.5.2. Proposition. The conic-rigidity matrix of a plane geometric graph G(p), non-collinear, has a maximum rank 31 VI - 6.
Proof. We show that the homogeneous system RC(G, p) x u'= 0 has a six-dimensional space of solutions generated by: T1 = (1,0,0, 1,0,0, ..., 1,0,0) T2 = (0,1,0, 0,1,0, .., 0,1,0) T3 = (0,0,1, 0,0,1, .., 0,0,1)
T4 = (2x1,Y1,0, 2x2,Y2,0, ..., 2xivi,yivi,0) T5 = (0,x1,2y1, O,x2,2Y2, ..., O,xivi,2y,vi) 2
2
2
2
2
2
T6 = (x1,x1 Yvy1, x2, x2Y2,Y2, ..., xlvl,xlvlYlvlYlvI).
The first three vectors, which correspond to the `translations' in 3-space, are
Walter Whiteley
26
clearly independent solutions. We check that the last three, which correspond
to `rotations', are solutions for a typical edge joining points (x1, y1) and (x2, y2) along the corresponding line
(Y2-Yt)x+(x1 -x2)Y+(x2Yt -x1 Y2)=0. For T4 we have
2x1(Y2 - yl)2 + (Y1)2(Y2 - Y1)(x1 - x2) + 0
- [2x2(Y1 - Y2)2 + (Y2)2(Y1 - Y2)(x2 - x1) + 01 2(Y2
- YO
X 1x1(Y2 - Y1) + Y1(x1 - x2) + x2(Y1 - Y2) + Y2(x2 - x1)1
2(Y2-YOM=0. A similar check works for T5. For T6 we have xi(Y2 - Yt)2 + 2x1 Yl(Y2 - Y1)(xt - x2) + YI(x1 -x2)2
-
Lx22(Y1
- Y2)2 + 2x2Y2(Yt - Y2)(x2 - x1) + Y2 (x2 - xt)2
_ (xi - x2)(Y2 -Yt)2 + (2x1 Yt - 2x2Y2)(Y2 - Yt)(x1 - x2)
+(Y1 -Y2)(xt -x2)2 =(Y2 -Yt)(x1 -x2) x [(x1 + x2)(Y2 - Y1) + (2x1 Yt - 2x2Y2) - (Yt + Y2)(xt - x2)] = (Y2 - Y1)(x1 - x2)(0) = 0.
In Example 1.5.1 we checked that these vectors were independent solutions for the three vertices of the triangle. The same check holds for any three non-collinear points. By analogy with spatial frameworks, we call a plane geometric graph with at least three vertices conic-rigid if the conic-rigidity matrix has this maximal rank 31 VI - 6. A single edge with distinct vertices is also conic-rigid, as is a single point with no edges. Example 1.5.1 shows that a non-collinear triangle
is conic-rigid. All the inductive techniques developed for frameworks in 3-space extend to conic-rigidity (see below, and Exercises 1.35, 1.36) and some techniques only conjectured for frameworks apply to conic-rigidity. (Exercise 1.37).
With these similarities in counts and in inductive techniques, we can ask if the conic-rigidity matrix RC(G, p) for a generic embedding in the plane, and the rigidity matrix R(G, q) for a generic embedding in 3-space, create the same matroid on the edges.
27
Matroids and Rigid Structures
1.5.3. Conjecture. A set E of edges is independent for generic embeddings of the graph as a bar framework in 3-space if and only if the set is conic-independent
for generic embeddings as a geometric graph in the plane.
If the rigidity matroid is maximal, as conjectured in the previous section, Conjecture 1.5.3 claims that the generic conic-rigidity matroid for a graph is also maximal. We have shown that a triangle is generically conic-rigid. We will show that the graph of any triangulated sphere is also generically conic-rigid, following the approach in section 1.4.
1.5.4. Vertex splitting theorem for conic-rigidity. Given a conic independent
geometric graph G(p) in the plane with a joint p1, and all incident bars (1, 2), (1, 3), (1, 4), .., (1, k), (1, k + 1), ..., (1, k + m) with p1- P2 not parallel to p1- p3, then for any k + m > 2, the new geometric graph on the vertex split on (1, 2), (1, 3) is conic-independent for almost all positions for the new vertex po.
Proof. We follow the proof of Theorem 1.4.6, choosing, as a limiting initial case, to add po at p1, with the `line' po, p1 assigned a direction D01 so that 2 2 2 2 = D12, D03 = D13, and Dot are independent 3-vectors. This creates the conic-rigidity matrix Do2
2
Dot
2
D 02
Dot 0
Do3
0
0
D12
2
2
2
0
0
-Do2
0
0
0
0
- D03
2 -D12
0
0 z
2
0 2
0
D13
0
-D13
0
0
D14
0
0
-D14
0
...
2
It is clear that this vertex adds rank 3 to the previous conic-rigidity matrix if and only if the three vectors DE21, D02 2, and D03 are independent. The remaining steps duplicate the steps of Theorem 1.4.6, creating the pattern of
a conic-rigidity matrix for the vertex split. Again this matrix remains independent if we move po to any nearby position with p1 - po = Dot. This completes the proof. Theorem 1.4.7 now extends immediately to conic-rigidity, since the proof was essentially topological.
28
Walter Whiteley
1.5.5. Proposition. For every graph of a triangulated sphere the geometric graph G(p) at a generic point p has rank 31 VI - 6 and is conic-independent.
What is the significance of these results? Consider a plane drawing of a 3-connected planar graph - with its associated faces. A CZ-spline assigns a quadratic function
z=A"x2+B"xy+C"y2+D"x+Ehy+F" for each face f" (except the exterior unbounded region). If two faces f", fi share an edge along the line Ljk: px + qy + r = 0 then the Ct requirement of a common tangent plane at all points on this edge becomes a simple algebraic condition on the functions: (A"x2 + B"xy + Chy2 + D"x + E"y + F")
- (A 'x2 + B'xy + Ciy2 + D'x + Eiy + Fi) = w"i(p2x2 + 2pqxy + q2y2 + 2prx + 2qry + r2). for some scalar w"i (see Exercise 1.38). Moreover, if we follow a cycle of faces
and edges around an interior vertex of the disc, the net difference is zero, giving the equation Y w"i(p2x2 + 2pqxy + q2y2 + 2prx + 2qry + r2) = 0
(sum over the vertex cycle).
Thus a CZ-spline corresponds to a choice for a quadratic function for an initial face, and set of scalars for the edges, satisfying these vertex conditions. (The complete proof reproduces the arguments of Lemma 1.3.4 and Theorem 1.3.5 for self-stresses on a plane framework.) These scalars look suspiciously like our conic-dependences, although these satisfy the six conditions: Y_ w"i(p2, 2pq, q2, 2pr, 2qr, r2) = (0, 0, 0, 0, 0, 0)
(sum over the vertex cycle).
We can use the orientation of the polyhedron to convert from scalars for a pair of adjacent faces to scalars for a pair of adjacent vertices. We must show that this extension from three equations to six is automatic
at any vertex. Given a point po = (xo, yo), and the three coordinates (A, B, C) = (p2, 2pq, q2) for a line px + qy + r = 0 through po, there is a unique 6-vector defined as follows:
(xo, yo, 1)#(A, B, C) = (A, B, C, -2Axo - Byo, -Bxo - 2Cyo, Axo + Bxoyo + Cyo) = (p2, 2pq, q2, 2pr, 2qr, r2) = L2.
Thus if Y; wij(Aij, Bid, C,;) = 0 then Lwi;(xi, yi, 1) # (Aid, Bid, Ci;) = 0. (The 6-vector
(A, B, C, -2Axo - Byo, -Bxo - 2Cyo, Axo + Bxoyo + Cyo)
Matroids and Rigid Structures
29
is interpreted as the coordinates of a quadratic function that is tangent to the xy-plane at the point p0.) This gives a set of scalars for all edges except the exterior boundary edges.
If the boundary is a non-degenerate triangle, then this triangle has rank 31 V- 61 = 3. This rank gives conic-rigidity - so the interior scalars extend to scalars on the boundary edges (see Exercise 1.38). The conic-dependence on the interior extends to a conic-dependence on the whole.
1.5.6. Proposition. For a triangulated disc in the plane, with a triangular boundary and a graph G(p), the space of CZ-splines with the zero quadric over a fixed triangle is isomorphic to the vector space of conic-dependences of G(p).
This gives, as a corollary, the basic theorem for the dimension of the space of CZ-splines.
1.5.7. Corollary. A generic triangulated plane disc with a triangular boundary has only trivial CZ-splines (all quadrics equal).
An analogous matroid exists for C3-splines, giving a matroid similar, but not identical, to the generic rigidity matroid in 4-space (see Exercises 1.40-1.42).
1.6. Scene Analysis of Polyhedral Pictures In section 1.3 we demonstrated a connection between polyhedral pictures
and the plane rigidity matroid. We shall now introduce a matroid that characterizes pictures of `general polyhedral scenes'. Basic similarities to rigidity, at the level of matroid theory, will again be evident. Consider an elementary plane picture formed by a set P of distinct points pj, and a set F of unordered sets of four distinct points f` = {Ph, P;, Pk, Pm} The points pj are lifted to heights zj, with the constraint that the four points of each face must remain coplanar in this spatial scene. For four non-collinear points, each face imposes one linear equation on the heights: Xh
xi
xk
xm
Yh
Y;
Yk
Ym
Zh
ZJ
Zk
Z.
1
1
1
1
=0 or
[p;pkPm]zh - [PJPkPm]zj- [PhP;Pk]Zm = 0
where the bracket [PjPkpm] is the corresponding minor of the determinant, recording the oriented area of the plane triangle PJPkpm. (If the four points are collinear, then a vertical plane will fit any selection of heights, so there is no condition.)
Walter Whiteley
30
We actually record a plane picture as an abstract hypergraph H = (V, F), with all faces as unordered quadruples of vertices, and an assignment P of plane points to the vertices. The constraints on the spatial scenes define an IFI x I VI homogeneous system M(H, P) x Z'= 0, and a corresponding matroid on the faces. Every picture whose points span the plane has a 3-space of trivial scenes - scenes with all points coplanar. Thus an independent set F of faces must satisfy IF'I 5 I V'I - 3 for every non-empty subset F'. Figure 1.16 shows some elementary plane pictures, with the faces shown as quadrilateral regions, using heavy lines between faces sharing two vertices. Figure 1.16.
(b)
(c)
1.6.1. Example. Consider the elementary scene in Figure 1.16a, with six faces and eight vertices. The matrix for this picture has the form: 10203041 -10103043
101P2P41
-LP1P2P31
0
0
0
0206071 -CPIP6P7J
0
0
0
10/02071
-CP1P:P6]
0
0
0
0
1020481
-102 3071
0
0 0
-10304031 0
10506P8]
- 106P7081
0
10307081
1_02P7P8J
0
0
10403081
104PS061
0
0
0
0
0
-1045081 [ 304P81
-10105061 10104P6J _10104P5J 0
10807081 -[P,07081
0
The matrix has rank 5 for generic pictures, giving a linear dependence and leaving only the trivial scenes with all vertices (and faces) coplanar. Even with one face removed, general pictures are independent, but have only trivial scenes. However, with two faces removed (any two), the matrix has rank 4 for a generic picture (Figure 1.16b), giving non-trivial scenes, with distinct planes for each face. Consider a special picture, such as Figure 1.16c, with the positions: Pi = (0, 0) Pz = Ps = (3, 3)
1)
P3 = (2, 1)
Pa = (3, 0)
P6 = (0,3)
p7 = (1, 2)
Ps = (2, 2).
(11
Matroids and Rigid Structures
31
The picture matrix for all six faces now has the form
-1 -1
3
-3
1
0
0
0
3
0
0
0
1
-3
0 0
0
-i
0
0
3
00 -1
0
0
-1 -3
9
0
0
-9 9 -9
0
0
0
0
0
-1
3
-3
1
1
0
1
0
1
This matrix has rank 4, giving non-trivial scenes with distinct planes for each face.
We want to characterize the generic elementary picture matroid, which occurs for general (e.g. algebraically independent) points in the plane. 1.6.2. Theorem. A hypergraph H of quadruples has realizations as an independent plane picture if and only if IF'I < I V'I - 3 for every non-empty subset of faces F'.
Proof. We already have seen that an independent plane picture satisfies IF'l < I V'I - 3 for every non-empty subset of faces F'. Assume that a hypergraph satisfies IFI S I V'I - 3 for every non-empty subset of faces F'.
We begin with the simpler transversal matroid, defined by the semimodular function g(F) = I VI. The transversal matroid is represented over the reals by the modified incidence matrix A(F): aid=tip
if
v,ef and aid=0
if
vjof,
for any set of algebraically independent reals tip.
By our assumption, F is also independent in this transversal matroid, giving independent rows in the transversal matrix. Since this matrix is IFI x I VI, the system A x X = 0 has a solution space of dimension at least 3.
Moreover, if we add three copies of any edge of F, we get a set F" that is still independent in the transversal matroid, so that we can choose a solution set of dimension 3 when restricted to the vertices of any edge. We write a basis for this chosen solution space as the IFI x 3 matrix Y, with rows Y = [xjyjw1], satisfying A x Y = 0. We can choose the matrix Y so that no values of wi are zero (take linear
combinations of the columns). If we divide each row Yj by the scalar wj, creating Y', and multiply the corresponding column Ai of A by wj, we obtain
another representation A' of the same matroid, with A' x Y' = 0 and Y; = [x; y;1]. We check that this matrix A' is the picture matrix up to multiplication of rows by non-zero constants. For a typical face of the form f = {v1, v2, v3, v4 }, the equation A' x Y' = 0 gives the equations
Walter Whiteley
32
t1x1 +t2x2+t3x3+t4x4=0 t1Y1 + t2Y2 + t3Y3 + t4Y4 =0
t11+t21+t31+t41=0 Given the matrix Y', these form three independent equations in the t;, for which the only solutions are scalar multiples of X2
X3
VX4
x1
x3
x4
tl = Y2
Y3
Y4
Y1
Y3
Y4
1
1
1
1
1
1
x1
x2
x4
x1
x2
x3
t3 = Y1
Y2
Y4
Y1
Y2
Y3
1
1
1
1
1
1
Therefore, the row of A' for edge f is: Q(10203047, -101P3041, 10102041, -10102031)-
If we take the picture with p. = (x1, y;), this row of A' is a multiple of the row of picture matrix M(F, P) for the edge f Thus an independent set of rows in A' corresponds to independent rows in M(F, P). We conclude that F has a realization as an independent elementary picture.
This is an elementary result for elementary pictures. A general plane picture involves faces with arbitrary (finite) sets of points which are constrained to be coplanar in a general spatial scene. There is an alternate system of equations that is traditionally used for these scenes. Each face f ` in the scene is recorded as a non-vertical plane a'x + b`y + z + c` = 0 or as a triple (a`, b`, c'). If vertex v j e f `, then we have the single linear equation in the variables z j, a`, b`, c`:
a`xj+b`yj+zj+c`=0 The corresponding system of equations creates a matroid on the rows of the coefficient matrix, or a matroid on the incidences I c V x F. A plane picture
S(P) is an incidence structure (or bipartite graph) S = (V, F; I) with an assignment P of plane points to the vertices V The 3-space of trivial scenes continues to exist, so a set of independent incidences satisfies II'I S I V'I + 31F'I - 3
on non-empty subsets F.
Matroids and Rigid Structures
33
Figure 1.17.
(a)
(C)
(b)
(d)
1.6.3. Example. We consider the picture in Figure 1.17a. This incidence structure of six vertices, three faces, and 12 incidences, creates the picture matrix 1
0
0
0
0
0
xt
Yi
1
0
0
0
0
0
0
0
1
0
0
0
0
x2
Y2
1
0
0
0
0
0
0
0
0
0
1
0
0
x4
Y4
1
0
0
0
0
0
0
0
0
0
0
1
0
x5
Y5
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
x2
Y2
1
0
0
0
0
0
1
0
0
0
0
0
0
x3
Y3
1
0
0
0
0
0
0
0
1
0
0
0
0
x5
Y5
1
0
0
0
0
0
0
0
0
1
0
0
0
x6
Y6
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
xt
Yt
1
0
0
1
0
0
0
0
0
0
0
0
0
x3
Y3
1
0
0
0
1
0
0
0
0
0
0
0
0
x4
Y4
1
0
0
0
0
0
1
0
0
0
0
0
0
x6
Y6
1
This has rank 12 = I VI + 3IF1- 3 for generic points - and the picture has only trivial scenes. For a special position, such as Figure 1.17b, with vertices Pt = (0, 0), P2 = (1, 1), P3 = (2, 0), P4 = (0, 3), P5 = (1, 2), P6 = (2, 3),
the matrix has rank 11- and has non-trivial scenes, as the three quadrilateral faces of a `triangular prism'. Each of these three faces has exactly four vertices - and we can return to the elementary picture matroid by row reducing on the final nine columns, provided no face has all vertices collinear:
Walter Whiteley
34 1
0
0
0
0
0
x, y,
1
0
0
0
0
0
0
0
1
0
0
0
0
x2
y2
1
0
0
0
0
0
0
0
0
0
1
0
0
x, y,
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
x2
y2
1
0
0
0
0
0
1
0
0
0
0
0
0
x2
y,
1
0
0
0
0
0
0
0
1
0
0
0
0
x, y,
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
x, y,
1
0
0
1
0
0
0
0
0
0
0
0
0
x, y,
1
0
0
0
0
0
1
0
0
0
0
0
0 X. y,
1
0
0
0
0
0
0
0
0
0
0
-[0203011
0
0
0
0
0
0
0
0
0
-[01030.]
0
0
0
0
0
0
0
0
0
020 .017
-[P1PJPs7
0
03P4P,1
[030106] 0
0
-10201061
-[P4sP6)
[0,02017 0
10,03067
-[P,P2Ps] [0203061 0
A similar reduction, applied to all picture matrices, is the foundation for the next theorem.
1.6.4. Theorem. The incidences I of an independence structure S = (V, F; I) are independent in a generic plane picture if and only if II'I S I V'I + 31F'I - 3 on all non-empty subsets I'.
Proof. For a generic picture, faces with less than four vertices impose no conditions, and will not change the independence of the structure. Faces of n > 4 vertices can be replaced by (n - 3) 4-valent faces, choosing three fixed vertices and adding the others one at a time. This converts the problem to a problem on an elementary plane picture H^(P), and the interested reader can check that the counts JI'J S I V'I + 3IF'I - 3 correspond to the counts F ^ < I V ^I - 3 on this elementary picture.
A more central matroid for computer science arises from one more truncation of the picture matroid. A sharp scene is a spatial scene in which any two faces lie in distinct planes. A sharp scene has a 4-space of trivial changes over the same picture: with one face fixed in the horizontal, we can change the vertical scale, in addition to the 3-space of scenes that change the plane of the selected initial face. The following corollary is used in computer algorithms for recognizing pictures of sharp scenes.
1.6.5. Corollary. For an incidence structure S = (V, F; I) a generic picture has sharp scenes if and only if II'I <, IV'I + 31F'I - 4 for all subsets I' with at least two faces.
Matroids and Rigid Structures
35
Figure 1.18 shows some generically sharp pictures (pictures with sharp scenes).
Figurel.18. III= IVI+31FI-4.(a)IFI=4,IVI=8,III=16;(b)IFI=9,IVI=13,111=36; (c) IFI=10, IVI=10, 111=36.
(a)
(b)
1.6.6. Example. To highlight the similarity to rigidity matroids, consider simple pictures in which each vertex is incident with exactly two faces. With this assumption, we have I V'i = 2I1'I on relevant subsets and the count for sharp scenes II'I < I V'I + 31F'I - 4 simplifies to the count I V'I < 31F'I - 4. This
assumption also means that the incidence structure describes a graph, or multi-graph - with `vertices' F and an `edge' for each element of V connecting its two faces.
If we row reduce the picture matrix on the columns for the vertices, the matrix assumes a rigidity-like pattern. For example, the matrix from Example 1.6.3 becomes 1
0
0
1
00 00 00 00
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
x,
y,
1
0
0
0
0
0
0
x2
y2
1
0
0
0
0
0
0
0
0
0
0
X3
y3
1
0
0
0
0
0
x4
y4
1
0
0
0
0
0
0
0
1
0
x5
y5
1
0
0
0
0
0
0
0
0
0
1
0
0
0
X6
Y6
1
0
0
0
0
0
0
0 0
x, y,
1
0
0
0
0
0
0
0 0
x2
y2
1
-x2 -y2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x4
y4
1
0
0
0
0
0 0
x5
y5
1
0
0
0
0
0
0
0
0
0
0
-1
x3
y3
1
0
0
0
-x5 -y5 -1 X6
Y6
-x, -y, -1
1
0
0
-x3 -y3
-x4 0
0
-1
-y4 -1 0
0
-x6 -y6 -1
The block in the lower right corner is a `3-frame matrix' analogous to the 2-frame used in section 1.2, and studied in the exercises. It does not directly
Walter Whiteley
36
correspond to a problem of rigid frameworks, although it does have a subtle interpretation in terms of bar and body frameworks in the plane (Exercise 1.45).
The picture matrix has an important geometric dual interpretation describing parallel redrawings of faces in space. Each vertex (a, b) in the picture corresponds to a normal (a, b, 1) of a non-vertical plane
ax+by+z+d=0. Each face containing (a, b) in the scene corresponds to a point (xo, yo, zo) in space satisfying the equation
axo+byo+zo+d=0. Thus the dual realization of the picture has a plane corresponding to each vertex and a point corresponding to each face. Different dual realizations have corresponding faces parallel. A trivial realization has all vertices at the same point, corresponding to a solution to the picture equation with all faces at the same place. A non-trivial realization has distinct vertices, with faces parallel to the original faces (same normal). A trivial realization has a three-dimensional space of trivial parallel redrawings
- the translations. A non-trivial realization has a four-dimensional space of trivial parallel redrawings generated by the translations and the dilations. 1.6.7. Example. Consider the dual of Example 1.6.6 (Figure 1.17c). This is, essentially, a triangular bipyramid: the top and bottom `vertices' are implicit in the intersection of three planes. For general choices for the normals of six
faces, the only realizations are trivial, with all vertices at the same point (Figure 1.17d). The `general' bipyramid has `general vertices' - but not general
face normals - giving a lower rank for the parallel redrawing matrix, and an extra parallel redrawing: the similarity map. Formally, this example has only three vertices, and six faces which meet in pairs along the three sides of a spatial triangle. As such, this becomes an
example of the parallel redrawing polymatroid on spatial graphs (see Exercises 1.54).
1.7. Special Positions Graphs which are isostatic (or independent) in generic realizations as a bar framework in the plane or in 3-space also have special critical forms that are dependent (or stressed) because of the underlying geometry of the framework.
Matroids and Rigid Structures
37
Similar critical forms arise for conic-rigidity and for plane pictures. There is some combinatorics, and a lot of geometry, in the study of these special positions.
1.7.1. Example. Recall the frameworks of Example 1.2.2. The general rigidity matrix has the form xi-x4 Y1-Y4
x4-XI Y4-Y2
0
0
0
0
x3-xi y5-y2 x6-xi y6-y5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
y1-y5
0
0
0
xi-Y6 Y1-Y6
0
0
x4-x2 y4-y2
0
0
0
0
0
0
0
0
x5-x2 Y5-y2
0
0
0
0
X2-X,
y2-y5
0
0
0
0
x6-X2 Y6-Y2
0
0
0
0
0
0
0
0
0
0
x4-x3 Y4-Y3
0
0
0
0
0
0
0
0
x5-x2 x6-x3
x2-x4 Y2-Y4
x3-x4 Y3-Y4
x2-x6 Yz-Y6
0
0
0
0
y,-y,
0
0
x3-x5
Y3-Y5
0
0
Y6-Y3
0
0
0
0
x3-x6 Y3-Y6
We previously saw that this has generic rank 9, but has low rank, corresponding to non-trivial infinitesimal motions, for six points on a conic section. A lower rank corresponds to all 9 x 9 minors being zero; thus a set of 84 conditions has to be checked. Each minor gives a polynomial equation in the coordinates of the six points: p(Pi5 P2, P3, P4, Ps, P6)=0
What do these polynomials look like? They are of degree 9 in the points. We have already seen that each polynomial is zero for six points on a conic section. This condition corresponds to an irreducible polynomial of degree 12 in the points (degree 2 in each point) that must be a factor of each of the minors. What is left in each minor after this polynomial has been factored out? Just a simple linear factor that turns out to be a trivial statement about distinct coordinates for the points. For example, if we delete the first two columns and the seventh column, the remaining minor can be calculated by Laplace expansion, using the first row. This gives the trivial factor (y, - y4) times a minor that must be (and is) the irreducible polynomial for the conic. If we delete the first two columns and the eighth column, the linear factor is (x, - x4). These inessential linear by-products of the deleted columns are called tie down factors, and a basic theorem is stated as follows. 1.7.2. Theorem. (White & Whiteley, 1983) For any graph G that is generically isostatic in the plane there is a polynomial pure condition CG(p,, ..., Piv), of degree val, -1 in each point of valence val, in the graph, such that all JEl x IEJ minors of the ridigity matrix are trivial linear factors times the pure condition.
Walter Whiteley
38
1.7.3. Example. We have seen that the graph of Figure 1.19a is generically isostatic in the plane (section 1.2), and, by Maxwell's Theorem of section 1.3, that this framework is dependent if and only if it is the projection of a spatial
configuration with at least two distinct faces. We found three conditions which guarantee such a projection: (i) (ii) (iii)
if the three points A, B, and C form a collinear triangle; if the three points A', B', and C' form a collinear triangle; if the three lines AA', BB', and CC' are concurrent.
These three conditions give three irreducible polynomial factors for the pure condition of the graph which are of total degree 12 in the points. Thus they
are the complete pure condition of the graph. This pure condition is also equivalent to the single geometric condition (which is algebraically reducible): the two triangles ABC and A'B'C' are perspective from a line (i.e. the usual conclusion of Desargues' theorem holds).
1.7.4. Example. In 3-space, the graph of an octahedron is generically isostatic
as a triangulated convex sphere (Figure 1.19b). A similar analysis of the minors shows that all AEI x JEl minors of the rigidity matrix are either simple
multiples of a basic pure condition that is of degree vale - 2 in each point, or are identically zero. A classical theorem shows that the pure condition is the projective condition that four opposite faces ABC, AB'C', A'BC', and A'B'C are on planes through a common point p (Figure 1.19c). Figure 1.19.
Since a rigidity matrix for these special positions represents a specialization
of the variables for a generic position, the corresponding matroid is a weak-map image of the generic rigidity matroid. The new circuits created in these specializations are geometric circuits rather than the generic circuits of the generic realizations (see Exercises 1.56, 1.57, and 1.58). As the examples have hinted, the critical forms for infinitesimal rigidity are given by projective geometric conditions. Engineers in the last century
realized that the statics and infinitesimal mechanics of a framework are
Matroids and Rigid Structures
39
invariant under projective transformations and Exercise 1.10 asks the reader to confirm that the rank of the plane rigidity matrix is projectively invariant. There is also projective geometry buried in the special positions of the spline
matroid, the scene analysis matroid, and the parallel redrawing matroid (Exercises 1.39, 1.51, and 1.53).
Example 1.7.1 and Theorem 1.7.2 indicated that single polynomial `pure conditions' control the lowering of the rank of matrix of a generic basis in the rigidity matrix. A similar analysis leading to polynomial `pure conditions' applies to the other matrices from splines, scene analysis, etc. There is a rich field of research in this interplay of geometry and combinatorics arising in the matroids of critical forms.
1.8. Conclusions The families of rigidity matroids, spline matroids, and scene analysis matroids
represent only the best known and more accessible examples of the large collection of matroids that arise in discrete applied geometry. A number of related matroids are presented in the exercises. We underline some common themes of these examples from discrete applied geometry.
(1) The matroid is defined by a sparse matrix. The non-zero entries correspond to `coordinates' of geometric objects in the real plane (3-space, etc.), and follow a combinatorial pattern based on a graph or incidence structure.
For sections 1.1-1.5, this incidence structure was the underlying graph. We then replaced each non-zero entry by a d-vector (drawn from the geometry) and each zero entry by the zero d-vector. The result
was an expanded pattern which passes through d copies of graphic matroid (with variables), and then an algebraic (or geometric) specialization.
Our task was to trace the effects of this specialization of the entries. More generally, as in section 1.6, we begin with an incidence matrix, and make vector substitutions under one `type' of vertex (e.g. the faces). In other applications, we even replace entries by rectangular matrices rather than single rows, creating a polymatroid. However, all of these examples carry properties inherited from the incidence matrix.
(2) Analysis of the matroid is based on the dual concepts of row dependences (self-stresses) and column dependences (infinitesimal motions).
(3) If variables are used for these entries, we find the `generic' matroid for the class of geometric realizations. Independent sets, and bases, in the generic matroid can often, but not always, be characterized by simple counts (semimodular functions) on the underlying combinatorial structures.
40
Walter Whiteley
(4) The cases where simple counts fail (rigidity in 3-space, conic-rigidity, etc.) point to a fundamental problem in defining matroids by semimodular functions that are negative on small sets. The inductive techniques being
developed represent matroidal approaches to a matroid problem. (5) There are fundamental similarities in apparently disparate geometric problems. These similarities become accessible in the patterns of the sparse matrices, and in the matroidal properties. In new examples, the patterns of the sparse matrix suggest the appropriate analog among our basic patterns, and permit an easy transfer of methods among the examples.
(6) The behaviour of non-generic examples creates polynomial conditions
that give interesting interactions of projective geometry, algebraic geometry, and combinatorics in the specialized matroids.
We conclude as we began. Engineering and computer science examples create a set of represented matroids for which the combinatorics of even generic examples are currently unsolved. The existing work applies, and illustrates, a number of matroid methods, including circuit elimination,
matroid union, and truncation. The examples also suggest some new approaches to represented matroids.
1.9. Historical Notes
The results of this chapter are drawn from a number of scattered papers stretching over the last 130 years. The explicit connections among the various concepts of rigidity appear in Gluck (1975) and Asimow & Roth (1978) and are presented in great detail in Connelly (1987). The explicit use of matroid methods for generic bar frameworks structures is drawn from Crapo (1985), Graver (1984), Lovasz & Yemini (1982), Recski (1984,1989), and Whiteley (1988a).
Theorem 1.2.3 is a combination of Laman's theorem (1970) and Lovasz & Yemini (1982). The proof follows Whiteley (1988a) and the 2-frame represents a matroid union of two copies of the graph. The inductive results in the last century are summarized in Henneberg (1911), and are developed in Tay & Whiteley (1985). The analysis of the semimodular functions in terms of matroid unions and truncations is related to classical matroid results of Edmonds (1970), Pym & Perfect (1970), and Brylawski (1985a, b).
Maxwell's theorem dates to the development of `graphical statics' by engineers and geometers in the last century (Maxwell, 1864; Cremona 1890).
The classical geometric approach is presented anew in Crapo & Whiteley (1988), along with the converse and its extensions. The approach to Maxwell's theorem and its converse used in section 1.2 is based on Whiteley (1982). The infinitesimal rigidity of convex triangulated spherical polyhedra dates
Matroids and Rigid Structures
41
back implicitly to Cauchy (1831), and explicitly to Dehn (1916). The generic results were extracted by Gluck (1975) and simplified in Graver (1984), and
in Tay & Whiteley (1985). The topological development of triangulated surfaces by vertex splitting is found in Barnette (1982), and greatly extended in Fogelsanger (1988). That vertex splitting preserves infinitesimal rigidity and conic-independence is found in Whiteley (1991a, 1991c). The problem of the generic dimension of the space of C2'-splines was raised by Strang (1973), and studied in Schumaker (1979) and in Alfeld (1987). The
basic result was proven in Billera (1986) and in Whiteley (1991b). The approach presented here is related to techniques of Chui & Wang (1983) and was developed by explicit analogy to spatial frameworks in Whiteley (1986c, 1990a, 1991 b).
The explicit matroid for general scene analysis was presented in Sugihara (1984), with a conjecture for Corollary 1.6.5. Whiteley (1988b) gave the first proof of the theorem, implicitly re-proving Crapo (1985) for the result on elementary pictures. Sugihara (1986) presents this proof anew, with a full background on the motivation and related problems in computer reconstruction of spatial objects. The dual parallel redrawing matroids were developed in Whiteley (1986a, 1988b). Parallel redrawing of spatial polyhedra and convex polytopes is basic to the study of Minkowski decomposition of polytopes (Shephard, 1963; Kallay, 1982; Smilansky, 1987).
Particular examples of geometric `critical forms' date back to the last century. The explicit study of `pure conditions' was carried out in White & Whiteley (1983, 1987). The general behaviour of bipartite frameworks appears for statics in Bolker & Roth (1980) and for infinitesimal mechanics in Whiteley (1984a). The projective invariance of statics and infinitesimal mechanics dates back at least to Rankine (1863), and is modernized in Roth & Whiteley (1981). The projective invariance of parallel redrawing appears in Kallay (1982) and is proven for CZ-splines in Whiteley (1986b, 1991b).
Exercises A difficult problem is indicated by *, an unsolved one by **. 1.1.
Show that for any graph G = (V, E) and any m > 0, the function f (E) = ml VI + k
defines a non-decreasing semimodular function on subsets of the edges. 1.2.
If, for k> -2m, we define circuits IE'I on the edges of a graph by I E'I = ml V'I + k + 1 and IE"I < ml V'l + k, verify directly that the circuit elimination
axiom for matroids is satisfied. Show that the semimodular function f(E) = IVI -1 defines the cycle matroid of a graph or multi-graph. 1.4. Show that the semimodular function f(E) = IVI defines the bicycle matroid of a graph. 1.5. (Whiteley, 1988a) Show that the semimodular function f (E) = 21 VI -1 defines 1.3.
42
Walter Whiteley
1.6.
the matroid union of the bicycle matroid and the cycle matroid of a graph. (This is the generic matroid for bar frameworks on the surface of a cone). Show that the frameworks in Figures 1.20b and 1.20c are each minimal rigid plane frameworks, i.e. removing any one bar leaves a finite motion. Conclude that `rigidity' does not induce a matroid on the plane framework in Figure 1.20a.
Figure 1.20.
(a)
1.7.
(C)
(b)
(a) Show that if G(p) and G(q) are two frameworks with the same bar lengths
for all edges, then p - q is an infinitesimal motion of the framework G((p + q)/2).
(b) Show that p - q is a non-trivial infinitesimal motion of G((p + q)/2) if G(p) is not congruent to G(q). (c) Give an example where G(p) is congruent to G(q) and p - q is a non-trivial infinitesimal motion of G((p + q)/2).
(d) Show that if G(p,) is a sequence of frameworks all with the same bar lengths but not congruent to p, such that limp; = p, then G(p) has a
1.8.
non-trivial infinitesimal motion. (e) (Local uniqueness) Show that if G(p) is infinitesimally rigid, then there is an open neighborhood N(p) in lg2iri such that if G(q) has the same edge lengths as G(p) and q e N(p), then G(q) is congruent to G(p). Show that an infinitesimal motion u of a non-collinear plane framework G(p) is non-trivial if and only if there is a pair of joints ph, pk (not joined by a bar) such that
(Ph-Pk)'(Ph-Pk)0. 1.9.
(Whiteley, 1984a) Recall that a plane conic centered at the origin (an ellipse, an hyperbola, or two lines) can be written [p1 ] [Q] [p1 ]` = r for a symmetric matrix [Q] and a constant r. Show that any bipartite graph KA,B, a framework with all its joints on such a conic has a non-trivial infinitesimal motion a ; = [a;] [Q] and
b, = [ba] [Q]
for a, e A and bj a B. Show that this is a non-trivial infinitesimal motion unless the conic contains all lines joining (ah, a;) and (by, bk).
Matroids and Rigid Structures
43
1.10. (Projective invariance) Show that if a non-singular projective transformation in the plane, T, takes the points p=(p1, ..., p,,i)tothepoints q=(T(pl), ..., T(pv)) then the bar frameworks G(p) and G(q) define the same matroid for any graph
G. (Hint: prove that T takes a dependent set to a dependent set, using self-stresses and the projective weights of the transformed points.) 1.11. (Crapo, 1988) Show that a graph G = (V, E) is generically isostatic in the plane if and only if the set E of edges is the disjoint union of three trees Ti such that each vertex of G is incident with exactly two of the trees Ti, and with the further property that no two distinct subtrees of the trees Ti have the same span.
1.12. Given a graph G = (V, E), the cone graph with vo is the graph G * vo formed by adding one new vertex vo having edges to all vertices of G (Figure 1.15a). Show, from the counts, that the cone graph G * vo is generically isostatic in the plane if and only if G is generically isostatic on the line. 1.13. (a) Show that adding a 2-valent joint to an isostatic plane framework gives an isostatic plane bar framework, if the new bars are not collinear. (b) Show that deleting a bar (1, 2) from an isostatic plane framework, and inserting a 3-valent joint po, with non-collinear bars (0, 1), (0, 2), and (0, 3) gives an isostatic plane bar framework for generic positions of po. (c) (Tay & Whiteley, 1985) Show that every generically 2-isostatic graph is generated from a single edge by a Henneberg 2-sequence of steps (a) and (b), and that every graph with such a sequence is generically 2-isostatic. 1.14. Give a Henneberg 2-sequence for each graph in Figure 1.4a. 1.15. Show that a set of edges E' is a circuit in the plane generic rigidity matroid if and only if IE'I = 21 V'I - 2 and 1E"I < 21 V"I - 3 for all proper subsets. *1.16. (Lovasz & Yemini, 1982) If a graph is 6-connected in a vertex sense, then it has realizations as an infinitesimally rigid plane bar framework. 1.17. (a) Show that all resolved loads on a plane framework are equilibrium loads.
(b) Show that for a plane framework with at least 2 distinct joints the equilibrium loads form a space of dimensions 21 VI - 3. * 1.18. (a) Extend the definition of polyhedra to 2-connected planar graphs. (b) (Crapo & Whiteley, 1988) Extend Theorems 1.3.3 and 1.3.5 to these 2-connected polyhedra. (c) Show that every self-stress on a plane framework is the sum of self-stresses on subframeworks with 2-connected graphs. (d) Conclude that a plane framework with a planar graph has a non-trivial self-stress if and only if it contains the projection of a spatial polyhedron with at least two distinct face planes. * 1.19. (a) (Crapo & Whiteley, 1988) Extend Theorem 1.3.3 to projections of arbitrary oriented polyhedra from 3-space.
(b) Show that all self-stresses from oriented projected polyhedra satisfy the cycle condition Y-wjk(Yj - Yk, Xk-Yj, Xjxk-XkYj)=(0, 0, 0) n
for each face-edge cycle in the oriented polyhedron. (c) (Crapo & Whiteley, 1988) Assume that a self-stress on the graph of an
Walter Whiteley
44
abstract oriented polyhedron satisfies the cycle condition: Xk - Xi, XjYk-XkYj)=(01 0, 0) n
for each face-edge cycle in the oriented polyhedron. Show that this framework is the projection of a spatial polyhedron, with distinct face planes over any edge with a non-zero scalar in the self-stress. (d) Give an example of the graph of a toroidal polyhedron with a self-stress
that violates the cycle condition and hence is not the projection of a toroidal polyhedron. 1.20. (a) (Whiteley, 1984a) Extend Exercise 1.9 to give an infinitesimal motion of any bipartite framework in 3-space with all joints on a quadric surface centered at the origin: [p1 ] [Q] [p1 ]` = r. (b) When is this infinitesimal motion non-trivial? 1.21. (a) A simple Henneberg 3-sequence is a sequence of graphs beginning with a triangle, and the following steps: (1) adding a 3-valent vertex; (2) removing one edge {1, 2} and adding a 4-valent vertex p0, with edges {0, 1}, {0, 2}, {0, 3}, and {0, 4}. Show that any graph with a simple Henneberg 3-sequence is generically 3-rigid.
(b) Show that the graph of an octahedron is generically 3-isostatic, by constructing a simple Henneberg 3-sequence. (c) Show that the bipartite graph K46 6 is generically 3-isostatic, by constructing a simple Henneberg 3-sequence.
1.22. (a) Show that the graph of Figure 1.9c is generically non-rigid in 3-space. (b) Show that this graph is a generic circuit in 3-space. 1.23. Show that the cone graph G * vo is generically 3-isostatic if and only if G is generically isostatic in the plane. 1.24. (a) Show that a 5-valent vertex, in an isostatic framework in 3-space with no four joints coplanar, can be replaced by one of the two patterns in Figure 1.11 a, producing smaller isostatic framework(s). (b) Give an example of a 5-valent vertex that requires the second replacement principle. (c) Give an example of a graph, with all vertices 5-valent, which is generically isostatic in 3-space.
1.25. (a) Give an example of a graph that is 11-connected in a vertex sense, but for which no spatial realizations are infinitesimally rigid in 3-space. ** (b) (Lovasz & Yemini, 1982) Show that if a graph is 12-connected in a vertex
sense, then it has realizations as an infinitesimally rigid bar framework in 3-space.
**1.26. (a) Show that the graph of any triangulated sphere has realizations as an isostatic framework with all joints on the unit sphere. (b) Characterize the graphs that give isostatic frameworks for some realization
with joints on the unit sphere.
*1.27. (Whiteley, 1987) Show that the graph of a triangulated sphere that is 4-connected in a vertex sense, plus one edge, is a circuit in the generic rigidity
Matroids and Rigid Structures
45
matroid for 3-space (Figure 1.15b). (Hint: use vertex splitting.) * 1.28. (Fogelsanger, 1988) Show that the graph of a triangulated surface in 3-space spans the generic rigidity matroid for its vertices in 3-space. 1.29. Give the plane analog of the vertex splitting theorem 1.4.6. 1.30. (a) Give the definition of infinitesimal rigidity for a bar framework in d-space. Show that an isostatic bar framework G(p) in d-space with I VI > d satisfies IEI = dl VI -
d(d + 1) 2
and
IE'I 5 dIV'J
- d(d2+ 1)
for all subsets I V'I > d. (b) Give the definition of static rigidity for a bar framework in d-space. Show
that infinitesimal rigidity and static rigidity are equivalent for bar frameworks in d-space. (c) Give an example, for each d > 2, of a framework that is not generically rigid, but satisfies IEI = dI VI - d(d + 1)/2 and E'I < dl V'I - d(d + 1)/2 for all subsets I V'I >, d.
1.31. (Bolker & Roth, 1980) Show that generic realizations of the graph K7.7 in 4-space are not infinitesmially rigid. (Hint: 14 points lie on a quadric in 4-space.) What is the dimension of its space of self-stresses? *1.32. (a) Show that the cone framework G*vo(p) is infinitesimally rigid in d-space if and only if the central projection G(II(p)) from apex po is infinitesimally rigid in (d - 1)-space. 1.33. A simplicial 4-manifold is a collection of tetrahedra such that each triangular face is shared by at most two tetrahedra, and the link of each vertex vo (the edges that form a triangular face of a tetrahedron with vo) forms a triangulated
2-sphere. The manifold is strongly connected if each pair of tetrahedra is connected by a path of tetrahedra and shared triangles. Show that a strongly connected 2-manifold satisfies the lower bound (Barnette, 1973)
IEI>4IVI-10 by the following sequence of steps (Kalai, 1987):
(a) The star of each vertex (the edges of triangles sharing this vertex) is generically rigid in 3-space. (Hint: use Exercise 1.32.) (b) If two infinitesimally rigid frameworks in 4-space share four vertices, their union is also infinitesimally rigid.
(c) The graph of a strongly connected simplicial 4-polytope is generically rigid in 4-space. 1.34. (a) (Gromov, 1986; Kalai, 1987) A simplicial 4-pseudomanifold allows the vertex link to be any triangulated 2-surface. Use Exercise 1.28 to extend the results of Exercise 1.33 to strongly connected pseudo-manifolds. (b) (Barnette, 1973; Kalai, 1987) Generalize Exercise 1.33 to give the lower bound
Walter Whiteley
46
IEI%dIVJ
- d(d 2+ 1)
for strongly connected simplicial d-manifolds. 1.35. (a) Show that if a conic-independent geometric graph G(p) in the plane has three joints p1, p2, and p3, then inserting a 3-valent joint po with no two of the lines (po, PO, (Po, P2), or (po, p3) parallel creates a conic-independent
geometric graph. (b) Conversely, given a conic-independent geometric graph with a 3-valent vertex, show that deleting this joint with its bars leaves a conic-independent geometric graph.
1.36. (a) Show that if a conic-independent geometric graph has four joints p p2, p3, and p4, of which no three are collinear, and an edge (p1, p4), then removing this edge and inserting a vertex po with edges (po, Pi), (POI P2),
(POI P3), and (po, p4) creates a conic-independent geometric graph for almost all positions of po. (b) Conversely, given a conic-independent geometric graph with a 4-valent vertex po, show that there is an edge (pi, pi) among the vertices adjacent to po such that deleting po and inserting (pi, p) leaves a conic-independent geometric graph. 1.37. Prove that if a conic-rigid (independent) geometric graph G(p) contains two disjoint edges (1, 2) and (3, 4) and an additional vertex v5, then the geometric graph with edges (1, 2), (3, 4) removed and an added point po joined by edges (0, 1), (0, 2), (0, 3), (0, 4), and (0, 5) is conic-rigid (independent) for generic positions po. 1.38. (a) (Chui & Wang, 1983) Prove that two quadratic surfaces
z=(A"x2+B"xy+C"y2+D"x+E"y+F") and
z=(A'x2+B`xy+C'y2+D'x+E'y+F') meet over a line px + qy + r = 0, with common tangent planes, if and only if
(A"x2+B"xy+D''x+E"y+F")-(A'x2+B'xy+C'y2+Dx+E'y+F) = w"'(p2x2 + 2pqxy + g2y2 + 2prx + 2qry + r2).
(b) Given any set of scalars (oii assigned to the interior edges of a triangulated plane disc, such that at each interior vertex Y wii(Aii, B,i, Cii) = 0,
i show that there are unique scalars on the three edges of the exterior triangle such that Y-i wii(Aii, Bii, Cii) = 0 at the exterior vertices as well.
1.39. (Projective invariance of conic-rigidity; Whiteley, 1986b) Show that if a non-singular projective transformation in the plane, T, takes the points p = (p1, .. p, v) to the points q = (T(pl ), ..., T(piv)) then the geometric graphs
G(p) and G(q) define the same conic-rigidity matroid for any graph G. 1.40. For a plane geometric graph G(p) define the triple-line vector for a directed edge (i, j) as
Matroids and Rigid Structures
47
D = (P j, 3P2 gii, 3Piiq j, q) = ((yi - yi)3, 3(yi - yi)2(xi - x), 3(yi - yi)(x, - xi)2, (xi - xi)3)
for i <j
and
D = -D
for i <j.
Define RCUB(G, p) as the - JEl x 41 VI cubic-rigidity matrix for a geometric graph with the following row for an edge (i, j):
[000
...
D
...
-Dj
...
000]
(a) Prove that the solution space to RCUB(G, p) x u= 0 has dimension at least 10. (b) Show that the graph of a tetrahedron is cubic-rigid for almost all plane geometric graphs G(p).
(c) Show that adding a general position 4-valent vertex to a cubic-rigid geometric graph produces a new cubic-rigid geometric graph. (d) Conclude that for the generic geometric graph in the plane, the cubic-rigidity matrix defines a matroid of rank 41 VI - 10.
(e) Show that dependences in the cubic-rigidity matroid correspond to C3-splines over a planar drawing of a 2-connected graph with triangular boundary. 1.41. (a) Prove that if a cubic-rigid (independent) geometric graph G(p) contains two disjoint edges (1, 2) and (3, 4) and two additional vertice v5, v6, then the geometric graph with edges (1, 2), (3, 4) removed and with an added point po joined by edges (0, 1), (0, 2), (0, 3), (0, 4), (0, 5), and (0, 6) is cubic-rigid (independent) for generic positions p0. (b) (Maehara, 1988) Show that the graph K677 comes from the graph K6 minus one edge, by a sequence of double-edges replacements of type (a). (c) Show that K6,7 is generically cubic-rigid. (d) Show, by using a quadric surface in 4-space, that K67 7 is not generically rigid in 4-space.
(e) Prove that the generic cubic-rigidity matroid and the generic 4-space rigidity are different.
** (f) Prove that each set of edges that is independent in the generic 4-space rigidity is independent in the generic cubic-rigidity matroid. 1.42. (a) Define the d -fold line vector for a directed edge (i, j) of a plane geometric graph G(p), and the corresponding IIEI x (d + 1)1 VI d -fold-rigidity matrix for a geometric graph. Show that for the generic geometric graph KIV (p) in the plane, with I VI > d, the d-fold-rigidity matrix defines a matroid of rank (d + 1) I VI - (d + 2)(d + 1)/2. (b) Prove that the generic d-fold-rigidity matroid and the generic (d + 1)-space rigidity are different, for d > 3.
(c) Show that dependences in the d-fold-rigidity matroid correspond to Cd +'-splines over a planar drawing of a 2-connected graph with a triangular boundary. 1.43. (White & Whiteley, 1987) A k frame G(d) is a multi-graph G = (V, E) (without loops) and an assignment d of vectors d,c- 08" to the edges. An infinitesimal
Walter Whiteley
48
motion of a k-frame is an assignment u of centers u, a 1k to the vertices such
that for each edge e joining v; and vj, de (ui - u) = 0. A k-frame is infinitesimally rigid if every infinitesimal motion is trivial, with all u; = c for a fixed vector c. A k-frame G(d) is isostatic if G(d) is a minimal infinitesimally rigid k-frame on its vertices (i.e. JEl = kI VI - k and G(d) is infinitesimally rigid).
Prove that for a multi-graph G, the following are equivalent: (1) G(d) is an isostatic k-frame for some d e 08kIEJ;
(2) G is the union of k edge-disjoint spanning trees; (3) G satisfies JEl = kI VI - k and IE'I S kI V'I - k for all non-empty subsets E'.
1.44. (Whitely, 1988a) Given a graph and a partition of the vertices (..., V, ...) of a multi-graph, the contracted graph G*(V*, E*) identifies all vertices of each partition class V and removes all loops. (a) Prove that a multi-graph has realizations as an infinitesimally rigid k-frame if and only if IE*l > kI V*J - k for all contracted graphs G* of partitions of the vertices. (b) Show that a multi-graph G that is 2k-connected in an edge sense is rigid as a generic k-frame.
1.45. A non-zero 3-vector L'= (Li, L,*, L3) represents a line in the plane: Li x + L'2 y + L3 = 0. A bar and body framework in the plane is a multi-graph
G, and an assignment L of lines L,i to the edges of the graph. The rigidity matrix for the framework is the rigidity matric for the corresponding 3-frame. (Visualize the lines as bars joining large rigid bodies for the vertices joined by bars along the lines of the edges.) (a) Prove that for a multi-graph G, the following conditions are equivalent:
(1) G(L) is an isostatic bar and body framework in the plane for some L e OB31EI;
(2) G is the union of three edge-disjoint spanning trees; (3) G satisfies IEI = 31 VI - 3 and IE'I S 31 V'J - 3 for all non-empty subsets E'.
(b) Show that a multi-graph G that is 6-connected in an edge sense has realizations as an infinitesimally rigid bar and body framework in the plane.
(c) (Whiteley, 1984c) Give an interpretation of the reduced simple picture matrix as a bar and body framework in the plane. 1.46. A 6-vector L= (L;, L2, L3, L4, L`5, L6) represents a line in 3-space if and only if Li L'4 + L'2 '5 + L3 L'6 = 0. (These are the Pliicker coordinates of the
line.) A bar and body framework in 3-space is a multi-graph G, and an assignment L of lines L;j to the edges of the graph. The rigidity matrix for the framework is the rigidity matrix for the corresponding 6-frame. (a) (Tay, 1984) Prove that for a multi-graph G, the following conditions are equivalent: jE (1) G(L) is an isostatic bar and body framework in 3-space for some L e l6;
(2) G is the union of 6 edge-disjoint spanning trees; (3) (Tay, 1984) G satisfies IEI = 61 VI - 6 and IE'I 5 61 V'J - 6 for all nonempty subsets F. (b) Give an example of a multi-graph G that is 11-connected in an edge sense, but has no realizations as an infinitesimally rigid bar and body framework
Matroids and Rigid Structures
49
in 3-space. 1.47. Define a body and hinge framework in 3-space as a graph G and an assignment
of hinge lines H" to the edges of G. An infinitesimal motion of a body and hinge framework is an assignment of screwSicenters S' = (S; , S' , S3, S;, S5, S6)
to the vertices such that, for each hinge, - S' = w''H" for a scalar wu. A body and hinge framework is infinitesmially rigid if the only infinitesimal motions are trivial, with all S' the same. (a) Show that a hinge between two rigid bodies in 3-space is equivalent to five bars passing through the line of the hinge. (b) (Tay & Whiteley, 1983; Whiteley, 1988a) Show that a body and hinge framework in 3-space is independent if and only if 51E'l < 61 V'1- 6 for all non-empty subsets of edges.
(c) (Tay & Whiteley, 1983; Whiteley, 1988a) Show that a body and hinge framework in 3-space is generically infinitesimally rigid if and only if 51E*I > 61 V*1- 6 for all contracted graphs G* from partitions of the vertices.
1.48. (a) Show that the faces and edges of a spherical polyhedron define a graph that is infinitesimally rigid as a generic body and hinge framework in 3-space.
(b) (Crapo & Whiteley, 1982) Show that the faces and edges of a spherical polyhedron form an infinitesimally flexible body and hinge framework in 3-space if and only if the edges and vertices form a bar framework with a non-trivial self-stress. 1.49. (a) Show that a multi-graph G = IF, E) has realizations as an independent picture of a simple picture if and only if the graph, with any added edge, is the union of three edge-disjoint spanning trees. (b) Show that a multi-graph G = IF, E) has realizations as an independent picture of a sharp simple picture if and only if the graph is the union of four edge-disjoint trees, and no three subtrees span the same vertices. 1.50. (a) Extend the definitions of section 1.6 to elementary pictures, and general pictures in (d -1)-space of scenes in d-space.
(b) Show that a hypergraph H of (d + 1)-tuples has realizations as an independent (d -1)-picture if and only if IF'I < I V'I - d for every non-empty
subset of faces F. (c) Show that a generic picture in (d -1)-space of an incidence structure S = (V, F; I) has sharp scenes in d-space if and only if II'I < I V'I + d I F'J - (d + 1)
for all subsets I' with at least two faces. 1.51. (Projective invariance of scene analysis) Show that the rank of the picture matrix is unchanged by a projective transformation of the picture.
1.52. (a) Show that for simple scenes in the plane, the independent sets are characterized by IE'I < 21 V'I - 3 for all non-empty subsets. (b) Show that a plane geometric graph G(p) at a generic point p is independent for parallel redrawings if and only if IEI = 21 VI - 3
and IE'I < 21 V'1- 3
for all proper subsets.
(c) Give an isomorphism between parallel redrawings of a plane geometric graph and infinitesimal motions of the corresponding bar framework, which takes trivial parallel redrawings to trivial infinitesimal motions.
50
Walter Whiteley
1.53. (a) For a configuration of points and planes in space, satisfying a fixed incidence structure S = (V, F; I), define the parallel redrawing matrix. (b) (Projective invariance of parallel redrawing; Kallay, 1982) Show that the
rank of the parallel redrawing matrix in 3-space is unchanged by a projective transformation of the points and planes of a configuration. 1.54. (a) (Whiteley, 1986a) Describe the parallel redrawing polymatroid for geometric
graphs in 3-space, and characterize the independent sets for generic geometric graphs. (b) (Whiteley, 1986a) Show that a non-trivial parallel redrawing of a geometric graph in 3-space induces a set of non-trivial infinitesimal motions of the corresponding bar framework. (c) Give an example of a geometric graph in 3-space with only trivial parallel drawings, but with non-trivial infinitesimal motions as a bar framework.
(d) (Shephard, 1963) Show that the graph of a triangulated spherical polyhedron has only trivial parallel redrawings in 3-space. (e) (Shephard, 1963) Generalize this to graphs of spherical polyhedra where any two vertices are joined by a path of triangular faces and edges. 1.55. (a) (Whiteley, 1990b) Choose an incidence structure I c V x F of vertices and faces that must be satisfied by spatial points and planes:
A"xi+B"yj+zj+C"=0. For each spatial realization, this creates a linear constraint on local changes to the face planes and the vertex points: A"(xj)' + B"(yj)' + (zj)' + (A")'xj + (B")'yj + (C")' = 0. Over the set of incidences, this creates an III x (31 VI + 3IFI) matrix RI(Q, q). Show that for the incidences of a spherical polyhedron, and any realization
with all faces and vertices distinct, the rows of RI(Q, q) are independent. 1.56. (a) Show that a collinear triangle is a geometric circuit for infinitesimal plane rigidity. (b) A simple plane framework is a plane framework which can be created from a single bar by adding a sequence of 2-valent joints, with the two added
bars not collinear. Show that a simple plane framework creates an independent rigidity matrix. (c) Show that the plane framework on K33 3 is a geometric circuit if the six points lie on a plane conic section, with no three collinear. (d) Show that the projection of a triangular prism in 3-space, with all faces distinct non-vertical planes, is a geometric circuit as a plane framework. 1.57. Give the geometric conditions for frameworks on each of the graphs in Figure 1.21 to be geometric circuits. 1.58. (a) Show that for the graph of an octahedron with aa', bb', cc' not bars, the spatial framework G(p) is dependent if and only if the six lines ab', b'c, ca', a'b, bc', and c'a lie on a projective line complex in 3-space. (b) Show that for the graph of an octahedron, the spatial framework G(p) is
dependent if and only if the four planes abc, a'b'c, a'bc', and ab'c' are concurrent in a point d.
Matroids and Rigid Structures
51
Figure 1.21.
(a)
(b)
(c)
(c) (Whiteley, 1986b) Show that for the graph of an octahedron, the plane geometric graph G(p) is conic-dependent if and only if the six lines ab', b'c, ca', a'b, bc', and c'a are tangent to a conic section. References Alfeld, P. (1987). A case study of multivariate piecewise polynomials, in G. Farin (ed.), Geometric Modeling: Algorithms and New Trends, pp. 149-60. SIAM Publications. Asimow, L. & Roth, B. (1978). Rigidity of graphs, Trans. Amer. Math. Soc. 245, 279-89. Barnette, D. (1973). A proof of the lower bound conjecture for convex polytopes, Pacific J. Math. 46, 349-54. Barnette, D. (1982). Generating the triangulations of the projective plane, J. Comb. Theory Ser. B 33, 22-230. Billera, L. (1986). Homology theory of smooth splines: generic triangulations and a conjecture of Strang, Trans. Amer. Math. Soc. 310, 325-40. Bolker, E. & Crapo, H. (1977). Bracing rectangular frameworks I, SIAM J. Appl. Math. 36, 473-90. Bolker, E. & Roth, B. (1980). When is a bipartite graph a rigid framework?, Pacific J. Math. 90,27-44. Brylawski, T. (1985a). Coordinatizing the Dilworth truncation, in L. Lovasz & A. Recski (eds), Matroid Theory, (Szeged, 1982), pp. 61-95. North-Holland, New York. Brylawski, T. (1985b). Matroid constructions, in N. White (ed.), Matroid Theory, Cambridge University Press. Cauchy, A. (1831). Deuxieme memoire sur les polygons et les polyedres, J. Ecole Polytechnique XVI, 87-98. Chui, C. K. & Wang, R. H. (1983). On smooth multivariate spline functions. Math. of Comp. 41, 131-42. Connelly, R. (1987). Basic concepts of rigidity, preprint chapter, Cornell University, Ithaca, New York. Crapo, H. (1985). The combinatorial theory of structures, in L. Lovasz & A. Recski (eds), Matroid Theory (Szeged, 1982), pp. 107-213, North-Holland, New York. Crapo, H. (1988). On the generic rigidity of plane frameworks, preprint, INRIA B.P. 105, 78153 Le Chesnay Cedex, France. Crapo, H. & Whiteley, W. (1982). Stresses on frameworks and motions of panel structures: a projective geometric introduction, Structural Topology 6, 42-82. Crapo, H. & Whiteley, W. (1988). Plane stresses and projected polyhedra I: The basic pattern, preprint, Champlain Regional College, 900 Riverside Drive, St Lambert, Quebec, J4P-3P2. Cremona, L. (1890). Graphical Statics (English translation), Oxford University Press. Dehn, M. (1916). Ober die Starrheit konvexer Poleder, Math. Ann. 77, 466-73. Edmonds, J. (1970). Submodular functions, matroids and certain polyhedra, in R. Guy et al. (eds), Combinatorial Structures and their Applications, pp. 69-87, Gordon & Breach, New York. Fogelsanger, A. (1988). Generic rigidity of triangulated homology 2-cycles, Ph.D. Thesis, Cornell University, Ithaca, New York.
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Walter Whiteley
Gluck, H. (1975). Almost all simply connected surfaces are rigid, in Geometric Topology, Lecture Notes in Mathematics 438, pp. 225-39. Springer-Verlag, Berlin and New York. Graver, J. (1984). A combinatorial approach to infinitesimal rigidity, preprint, Department of Mathematics, Syracuse University, Syracuse, New York 13210. Gromov, M. (1986). Partial Differential Relations, Springer, Berlin, Heidelberg, and New York. Henneberg, L. (1911). Die Graphische Statik, Leipzig. Imai, H. (1985). On the combinatorial structure of line drawings of polyhedra, Discrete Appl. Math. 10, 79-82. Kalai, G. (1987). Rigidity and the lower bound theorem I. Invent. Math. 88, 125-51. Kallay, M. (1982). Indecomposible polytopes, Israel J. Math. 41, 235-43. Laman, G. (1970). On graphs and the rigidity of plane skeletal structures, J. Engineering Math. 4, 331-40. Lovasz, L. & Yemini, Y. (1982). On generic rigidity in the plane, SIAM J. Alg. Discrete Meth. 3, 91-8. Maehara, H. (1988). On Graver's Conjecture, Ryuku University, Nishihara, Okinawa, Japan. Maxwell, J. C. (1864). On reciprocal diagrams and diagrams of forces. Phil. Mag. (4) 27, 250-61. Pym, J. S. & Perfect, H. (1970). Submodular functions and independence structures, J. Math. Analysis Appl. 30, 1-30. Rankine, W. J. M. (1863). On the application of barycentric perspective to the transformation of structures. Phil. Mag. (4) 26, 387-8. Recski, A. (1984). A network approach to rigidity of skeletal structures II, Discrete App!. Math., 8, 63-8. Recski, A. (1989). Matroid Theory and its Applications, Springer-Verlag, Berlin, and Akademiai Kiado, Budapest. Roth, B. & Whiteley, W. (1981). Tensegrity frameworks, Trans. Amer. Math. Soc. 265, 419-45. Schumaker, L. L. (1979). On the dimension of spaces of piecewise polynomials in two variables, in W. Schemp & K. Zeller (eds), Multivariate Approximation Theory, pp. 396-412. Birkauser, Basel. Shephard, G. C. (1963). Decomposible convex polyhedra, Mathematika 10, 89-95. Smilansky, Z. (1987). Decomposibility of polytopes and polyhedra, Geom. Dedicata 24, 29-49. Strang, G. (1973). The dimension of piecewise polynomial spaces and one-sided approximations, in Proceedings of the Conference on Numerical Solution of Differential Equations (Dundee, 1973), Lecture Notes in Mathematics 365, pp. 144-52. Springer-Verlag, New York. Sugihara, K. (1984). An algebraic and combinatorial approach to the analysis of line drawings of polyhedra, Discrete Appl. Math. 9, 77-104. Sugihara, K. (1985). Detection of structural inconsistency in systems of equations with degrees of freedom, Discrete Appl. Math. 10, 297-312. Sugihara, K. (1986). Machine Interpretation of Line Drawings, M.I.T. Press, Cambridge, Mass. Tay, T: S. (1984). Rigidity of multi-graphs I: Linking rigid bodies in n-space, J. Comb. Theory Ser. B 36, 95-112. Tay, T: S. (1991). On generically dependent bar and joint frameworks in space, in Structural Topology (to appear). Tay, T: S. & Whiteley, W. (1983). Recent progress in the generic rigidity of frameworks, Structural Topology 9, 31-8. Tay, T: S. & Whiteley, W. (1985). Generating isostatic frameworks, Structural Topology 11, 21-69. White, N. & Whiteley, W. (1983). The algebraic geometry of stresses in frameworks, SIAM J. Alg. Discrete Meth. 4, 481-511. White, N. & Whiteley, W. (1987). The algebraic geometry of motions in bar and body frameworks, SIAM J. Alg. Discrete Meth., 8, 1-32. Whiteley, W. (1982). Motions, stresses and projected polyhedra, Structural Topology 7, 13-38. Whiteley, W. (1983). Cones, infinity and one-story buildings, Structural Topology 8, 53-70. Whiteley, W. (1984a). Infinitesimal motions of a bipartite framework, Pacific J. Math. 110, 233-55. Whiteley, W. (1984b). Infinitesimally rigid polyhedra I: Statics of frameworks, Trans. Amer. Math. Soc. 285, 431-65. Whiteley, W. (1984c). A correspondence between scene analysis and motions of frameworks,
Matroids and Rigid Structures
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Discrete Appl. Math. 9, 269-295. Whiteley, W. (1986a). Parallel redrawings of configurations in 3-space, preprint, Champlain Regional College, 900 Riverside Drive, St Lambert, Quebec J4P-3P2. Whiteley, W. (1986b). The analogy between algebraic splines and hinged panel structures, preprint, Champlain Regional College, 900 Riverside Drive, St Lambert, Quebec J4P-3P2. Whiteley, W. (1987). Infinitesimally rigid polyhedra II: Modified spherical frameworks, Trans. Amer. Math. Soc. 306, 115-39. Whiteley, W. (1988a). Unions of matroids and rigidity of frameworks. SIAM J. Discrete Math. 1, 237-55. Whiteley, W. (1988b). A matroid on hypergraphs, with applications to scene analysis and geometry, Discrete Comp. Geometry 4, 75-95. Whiteley, W. (1990a). The geometry of CZ splines, preprint, Champlain Regional College, 900 Riverside Drive, St Lambert, Quebec J4P-3P2. Whiteley, W. (1990b). Determination of spherical polyhedra, preprint, Champlain Regional College, 900 Riverside Drive, St Lambert, Quebec J4P-3P2. Whiteley, W. (1991a). Vertex splitting in isostatic frameworks, Structural Topology 16, 23-30. Whiteley, W. (1991b). The combinations of bivariate splines, in P. Gritzmann & B. Sturmfels (eds), Applied Geometry and Discrete Mathematics, The Victor Klee Feitschrift, AMS DIMACS Series, in press. Whiteley, W. (1991c). A matrix for splines, in Progress in Approximation Theory, Niven & Pinkus (eds), Academic Press, in press. Whiteley, W. (1991d). Weaving, sections and projections of polyhedra, Discrete Appl. Math., in press.
2
Perfect Matroid Designs M. DEZA
2.1. Introduction A perfect matroid design (PMD) is a matroid such that all flats of rank i have
the same cardinality a;. Some other names used for PMDs are: uniformly designed lattices, geometric lattices of triangular type, combinatorially homogeneous matroids, geometric designs, matroid designs, and Murty-Young-Edmonds designs (MYEDs). PMDs were introduced in Murty (1970) and the first study of them was given in Murty, Young & Edmonds (1970). The sequence a = (ao, al, ..., ar_ 1, ar) of flat sizes for a PMD of rank r is called its a-sequence. By the same process of geometrization
as in Brylawski (1986) p. 171, it is easy to obtain from any PMD with a-
sequence (ao, al, ..., a,_1, ar) a PMD with a-sequence (ao - ao = 0, al - ao) .. ., ar_ 1 - ao, ar - ao) and further a PMD with a-sequence
-0 0 a1-ao
C
al-ao-1 al - ao
ar-l - ao ar - OCO
al - ao
al - ao
i.e. a combinatorial geometry. Without loss of generality (as far as the problems considered in this chapter are concerned) we shall suppose throughout that ao = 0, al = 1. Though PMDs are a very special case of matroid a PMD is nevertheless a common generalization of three highly symmetric finite objects - finite geometries, designs, and commutative Moufang loops. Also, two more reasons
for being interested in PMDs come from combinatorics. One reason is that the existence of each PMD implies the existence of many designs, and it will be seen that Proposition 2.2.1 below can be a way to look for new ones. The other reason is that almost all PMDs correspond to the case of maximality for some inequality generalizing many results from coding theory and the extremal theory of finite sets (see Proposition 2.3.1 below). Given a PMD G(S) of rank r, let G(S):1 (which is called 1-truncation of
Perfect Matroid Designs
55
G(S)) denote the PMD of rank r - 1 having the set of all flats of G(S) except hyperplanes as the lattice of flats. By induction for any 1 < i < r - 1 define the i-truncation G(S): i of G(S) as the 1-truncation of G(S):(i - 1); define the 0-truncation G(S):0 as G(S) itself. We will call the contraction of the PMD G(S) on a fiat F' the PMD with the lattice {F I F' c F and F is a flat of G(S)} of flats; the contraction can be seen as `truncation from the bottom'. We will call the restriction (Brylawski, 1986, p. 131) of PMD G(S) on a flat F' the
PMD with lattice {F' r F I F is a flat of G(S)} of flats. Any i-truncation, 0 < i < r - 1, of a PMD, any contraction, and any restriction of a PMD on one of its flats are PMDs. The known examples of PMDs are: (1) i-truncations AG(d, q): i, 1 < i < r - 1, of affine geometries of dimension d (AG(d, q):0, i.e. AG(d, q) itself included) having rank d - i + 1 and a = (0, 1 = q°, q = ql, ..., t, qd); (2) i-truncations PG(d, q): i, 1 < i < r - 1, of projective geometries of qd-
dimension d (PG(d, q): 0, i.e. PG(d, q) itself included) having rank
d-i+1 and 0,1=q
1
q-1
q+I-_
q 2-1
q-1
a i_1
,..., q q-1
d+1 _1 q
q-1
;
(3) Steiner systems S(t, k, v) having a = (0, 1, 2, ..., t -1, k, v);
(4) PMDs of rank 4 with a = (0, 1, 3, 9, 3") arising from finite commutative Moufang (but not associative) loops; we will call them triffids (they are surveyed in section 2.5).
The special case S(k, k, v) of a Steiner system is a truncation of a Boolean
algebra. Any PMD whose a-sequence is the same as that of AG(d, q): i (d >, 3, q > 3) of PG(d, q): i (d > 3) or S(t, k, v) is isomorphic to AG(d, q): i or PG(d, q): i or S(t, k, v), respectively. For d > 3 AG(d, g) exists if and only if q is a power of a prime; the same
is true for PG(d, q), d > 3. The classification of all S(t, k, v) is a big open problem. The study of triffids started only recently. Obviously, it will be difficult to achieve the classification of all PMDs. The natural approach to this central problem will be, for example, the study of PMDs of rank 4 (because the rest is too hard) and, in particular, PMDs with a = (0, 1, 3, a3, a4) (those being analogs of S(3, k, v)) or PMDs with a = (0, 1, 3, 9, 3") (triffids and AG(d, 3):(d - 3)). This chapter contains five more sections. Section 2.2 gives some properties of the lattice of flats of a PMD, some parameters and links among them, some inequalities for parameters, and classification of extreme cases. Section 2.3 gives information about the possible properties of a hyperplane family
56
M. Deza
of being a `large' intersection family and a subscheme of a Johnson association
scheme. PMDs of rank 4 are considered in section 2.4. Section 2.5 gives a short survey on triffids. Section 2.6 mentions some generalizations and analogs of PMDs.
2.2. Some Lattice Properties and Parameters of PMDs In this section we consider the lattice L of all flats of a PMD. In particular, we give related designs, the Sperner property of L, some rank invariants (especially Whitney numbers), and all known necessary conditions for the existence of a PMD in terms of the a-sequence. A PMD of rank 3 is precisely a BIBD (balanced incomplete block design) with ,. = 1; in other terms, it is an S(2, k = a2, v = a3) or (b, v, r, k, 1)configuration. PMDs of higher rank correspond to very special designs. Propositions 2.2.1-2.2.3 below are from Murty, Young & Edmonds (1970). 2.2.1. Proposition. Given a PMD; then
(i) the set of all j -flats forms the set of blocks of a BIBD on the set of all 1 -flats, for 1 < j < r (i.e. a PMD is an MD; see section 2.6 below); (ii) the set of independent subsets of cardinality j is the set of blocks of a BIBD, for 1 < j < r; (iii) the set of circuits of cardinality j is the (possibly vacuous) set of blocks of a BIBD, for 2 <j < r + 1. Also, each of the three sets considered above is the set of blocks of some S(t, k, v), 2 < t < /3, where /3 is a maximal integer m such that every m-subset of S is independent.
The next property of the lattice of flats of a PMD considered here is its Sperner property, i.e. the property that the largest Sperner family (i.e. antichain) of this lattice is the set of all i-flats for some i, 1 < i 5 r - 1. This follows from a theorem of Baker given in Welsh (1976), Chapter 16.5.
Moreover, using Deza, Erdos & Frank] (1978) one can see that for a, sufficiently large (with respect to this largest antichain is exactly the set of all hyperplanes. Now we will consider some lower bounds for ISO = a2. Given a PMD G(S) of rank r with a = (0, 1, ..., a,) that is not S(k, k, v), denote a, by v and a,-, by k; let c be the minimum circuit cardinality of G(S). 2.2.2. Proposition. v >, k + min (k, c, r).
Any PMD, other than S(k, k, v), satisfies k >, r >, c and so has v >, k + c. We have k = c = r = v/2 for AG(3, 2) (it is S(3, 4, 8)) and for Witt design S(5, 6, 12).
Perfect Matroid Designs
57
The next property of a PMD is that for any i-flat F' and 1-flat F' with F' c F' the set F' - F' is partitioned by sets of the form F'+ 1 - F' where F'+ 1 is an (i + 1)-flat such that F' c F'+ 1 c F'. This is a general property of matroids (see Brylawski, 1986). Denote by t(i, j, 1) the number of j-flats F' such that F'= F' c F'; the following result comes directly from this property.
2.2.3. Proposition. The number t(i, j, 1) is independent of the choice of F', F'; moreover ai - ai
t(0, 1, 0 = ai, t(i, i + 1, 1) =
ai+ 1 - ai
y' r t(0,i,r)= j=1 ai - aj and
t(i, k, 1) = t(i, f, l)t(f j, l)/t(i, f j)
0,
for
The number t(0, i, r), i.e. the number of all i-flats, is the Whitney number i of the second kind of our PMD. Denote by L the lattice of all flats of G(S), by u(x, y) the Mobius function of the lattice, and by r(x) the rank of x e L. The Whitney numbers wJ 1), W.!2) of the first and second kind are defined by W!1) _
Y u(O, x)8(r - r(x), r -j), xcL
W! 2)
_ Y 5(r-r(x), r-j). xeL
Example 8.5.9 of White (1987) discusses log concavity of the sequence W'2),
i.e. the inequality W(2)> +(i
-W(2)
i+)I )
2.2.4. Proposition. (Brini, 1980) Let x, y e L and x c y; denote r(x) = i, r(y) = 1. Then µ(x,y)=(-1)i-i
i [t(i,j-I,j)-t(i+1,j-I,j)] j=i+1
So, the functions ai, t(i, k, 1), u(x, y) and both Whitney numbers are rank invariants; another rank invariant considered in Brini (1980) will be mentioned
at the end of the next section. This section gives necessary conditions for the existence of a PMD in many different terms. In terms of the a-sequence, which is our basic rank invariant, independent necessary conditions for the sequence (0, 1, a2, ..., a,) to be the a-sequence of a PMD are (Murty, Young & Edmonds, 1970):
(1)
fl
ai - aj
1
f=i+1 al-af-1
is a non-negative integer for 0
<j < I 5 r;
M. Deza
58
(2) (a;-ai_1)divides a;+1-aifor 2
An example is provided (R. Wilson, unpublished, indicated in Murty, Young & Edmonds, 1970) by the sequence (0, 1, 3, 7, 43); moreover, Wilson showed that any PMD G(S) with 2 > 3, (a3 - (X2) = (2 - 1)2 has the property
ar(a2 - 2) = (2 - Of - 1 for some f In Ray-Chaudhuri & Singhi (1977) the inequality ai - a; ? (a2 - 1}'- i(aj_ 1 - a; _ ) was proved for 2 < i <j < r, and the cases of equality were classified.
2.3. Intersection Properties of the Hyperplane Family Consider a PMD G(S) with hyperplane family Mr-' and parameters a = (aa = 0, a1 = 1, a2, ..., ar); denote ar = v, ar-1 = k, {ao, a1 , ..., ar-2} = T Denote by A(T, k, v) any set of k-subsets of a given v-set such that the size of any pairwise intersection is a member of the given set T of integers. Of course, Mr-1 is an A(T, k, v). We will give below information known about Mr-1 with respect to the following possible special properties of A(T, k, v)s: (1) a maximal A(T, k, v) is an A(T, k, v) that is not a proper subset of an A(T k, v);
(2) more specially, a largest A(T, k, v) is an A(T, k, v) with the largest number of k-sets; (3) a complete A(T, k, v) is an A(T, k, v) such that any number from T occurs
as the size of a pairwise intersection of k-sets;
(4) an A(T, k, v)-subscheme is an A(T, k, v) that is a subscheme of the Johnson association scheme of all k-subsets of a v-set.
Here, an association scheme on a finite set X with n classes (see, for example, Bose, 1963; Hamada & Tamari, 1975) is a partition X x X = No u N1 ... u N such that (1) (x, y) E No if x = y; (2) (x, y) E Ni iff (x, y) E N;; (3) for any x E X the number mi = I { y e X I (x, y) E Ni } I depends only on
i;
(4) for any x, y e X with (x, y) E Ni the number PPk = I {z e X I (x, y) E NN, (y, z) e Nk } I depends only on i, j, k.
We say that the pair (x, y) is i-associated if (x, y) E Ni. For n = 2 the pairs in each class can be seen as the edges of a strongly regular graph (introduced by Bose, 1963) and any such graph gives rise to an association scheme with two classes.
Perfect Matroid Designs
59
The Johnson scheme J(k, v) is defined by X = S(k, k, v) and (x, y) E Ni if
Ixnyl=k - i. A PMD G(S) can be characterized (if v > vo(k)) in the class of all A(T, k, v)s
by the following extremal property. 2.3.1. Proposition. (Deza, Erdos & Frankl,1978; Deza, 1978) Let A = A(T, k, v); then there exists a polynomial vo(T, k) of argument k such that v > vo(T, k) implies:
G) IAI ,
v-i' k-i icT v -1
AI _ (ii)
I
;
k-i
if and only if A is the set of all hyperplanes of some PMD with a = (ao, a 1, ..., cc).
For the special case T = {0, 1, 2, ..., r - 2} this proposition is just the Schonheim-Johnson bound for packings and codes (see, for example, Deza, 1978); for this T we have vo(T, k) = k and the corresponding PMD is a Steiner system S(t = r - 1, k, v). From a remark of P. Frankl (1983) it follows that vo(T, k) 5 (k - ai_ 2)2(k - ar_ 2 + 1) + k for the case T = {0, 1, 2, ..., r - 3, ar _ 2 }, i.e.
if G(S) is an `extension' of an S(r - 2, k, v). For the special case
T = {0, 1, 2, ..., r - 3, r - 1} of the above case Frankl gave a better bound vo(T, k) < (k - r + 1)2 + k; therefore vo(T, k) = k for all PMDs with a = {0, 1, 3, 9, 3"}, n > 4. The smallest vo(T, k) is unknown for truncations of PG(d, q) and AG(d, q) and for the triffid with a = {0, 1, 3, 9, 81}. Frankl (1983) has also shown that IA(T, k, v)I =O(vITI-1) unless there is a
PMD with a = (ao, al, ..., ar_ 1). Let us also note that using the family of elliptic quadrics over PG(d, q), q > 2, Deza, Frankl & Hirschfeld (1985) showed that for v > vo(q),
A({0, 1, 2,q+1}, q2+1,v)>cq 92
v
+ 1 )(
v-1 (v-2 v)
q
2
1
(
2
q -qq
1
),
where cq is a positive constant with c9- 1 as q- oo, i.e. the corresponding families come fairly close to PMDs. So Mr-' is a largest (and, in particular, maximal) A(T, k, v) for any v if G(S) is an S(t, k, v). M' ' is a maximal A(T, k, v) for any v if G(S) is some i-truncation of either affine or projective space; it follows from the more general Theorem 3a of Deza & Singhi (1980) that M' ' is a maximal A(T, k, v)
if v < k2/ar_2. Hence Mr-' is maximal whenever v is sufficiently `large' or `small' with respect to k. The hyperplane family Mr-' of a triffid (the only remaining example of a known PMD) is also maximal; the maximality was
M. Deza
60
proved in Theorem 3b of Deza & Singhi (1980) for a class of PMDs of rank
4 including any PMD with a = (0, 1, 3, k, v). So, we conjecture that the hyperplane family of any PMD with a = (0, 1, a2, ..., ar_ 2, k, v) is a maximal A(T, k, v).
Mr-1 can be a non-complete A(T, k, v) but all such cases in the class of all known examples of PMDs are classified as follows. 2.3.2. Proposition. Let r > 3. Then (i) the i-truncation of a projective geometry PG(d, q) has a complete M'_' if and only if i > (d + 1)/2; (ii) th i-truncation of an affine geometry AG(d, q) has a complete Mr-1 if and only if i > d/2; (iii) each triffid has a complete Mi(iv) the Steiner system S(t, k, v) has an incomplete M"-' only in the following cases (if an S(t, k, v) exists): (a) (t, k, v) = (2, n + 1, n2 + n + 1) and is a projective plane of order n; (b) t = 3 and (k, v) = (6, 22), (4, 8), (12, 112); (c) (t, k, v) = (4, 7, 23), (5, 8, 24);
(d) (t, k, v) = (t, t + 1, 2t + 3) and t + 3 is a prime number; (e) (t, k, v) = (t, t + 1, 2t + 2) and t + 2 is a prime number. The most difficult part, (iv), of the above proposition is given in Gross (1974);
parts (i) and (ii) are given in Deza & Singhi (1980). We note that a triffid with a = (0, 1, 3, 9, 34) is complete, but AG(4, 3) is not. Any PMD with v sufficiently large (with respect to k) has a complete Mr-1; more exactly, Proposition 2.3.1 implies the following.
2.3.3. Proposition. Let the PMD G(S) have an incomplete M'-' and, more aj for some hyperplanes H1, H2 and some j, 0 <j < r - 2. Then v < vo(T- {aj}, k).
exactly, IH1 n H2 1
Given an i-flatEand au-flatFsuch that EcF,0cF,0
c(i, u) = t(i, u, r) - Y t(i, w, u)c(w, u). «wsu
As an immediate corollary we have the following test of completeness for
M"-': it is complete if and only if all numbers c(i, r - 1), 0 < i < r - 1 are positive. From the above proposition one can see that
Perfect Matroid Designs
c(i,
61
i+1)=(v-ai)/(ai+1-ai)-1=(v-ai+1)/lai+1-ad >0
for any 0 < i < r - 2 and, in particular, c(r - 2, r - 1) > 0 always. Also, it is easy to check that c(r - 3, r - 1) = 0 if and only if the contraction of our PMD on a (r - 3)-flat is a projective plane; in particular, Gross (1974) gives that S(t, k, v) has c(r - 3, r - 1) = 0 if and only if (t, k, v) = (3, 6, 22), (4, 7, 23), (5, 8, 24), (3, 4, 8) or (3, 12, 112), if it exists. PMDs of rank 4 having c(1, 3) = 0 are classified (under the name of finite regular locally projective spaces of dimension 3) by Doyen & Hubaut (1971). Each of them is either a projective space, or an affine space, or a Lobachevski space (in terms of Doyen & Hubaut, 1971) of type az - a2 + 1 or a2 + 1, i.e.
each top rank 3 interval of the lattice of flats is a projective plane of order q = az or q = a2 + a2. Concerning PMDs of rank 4 having c(0, 3) = 0: there is a conjecture in Cameron (1980) that each of them is a truncation of a Boolean algebra or projective space of rank 4 or 5. The number c(i, u) was generalized by Brini (1980) in the following direction
- given an i-flat E and a u-flat F, such that E c F, 0
i < u < r, denote by c(i, j, u) the number of all j-flats F1 such that F1 n F = E. Of course, c(i, u) is just c(i, u, u). Then c(i, j, u) is y(r - 1, r -j, r - u) in the notation of Brini (1980); in that paper it was proved that c(i, j, v) is independent of the choice of E and F and, moreover, that the following holds. r-i
2.3.5. Proposition. c(i, j, u) = I (-1)p-it(i, p, u)t(p, j, r) p=r-u
p
fl
[t(i, q - 1, q)
q=i+1
- t(i + 1, q - 1, q)]. The remainder of this section will give information known about PMDs in which Mr-1 is a subscheme of a Johnson association scheme. Any PMD of rank 3 is exactly a BIBD with A = 1; its line graph is strongly regular and hence Mr-1 will be a subscheme. 2.3.6. Proposition. Let r > 3. Then (i) (Hamada & Tamari, 1975) any i-truncation of PG(d, q) is a subscheme; (ii) (Hamada & Tamari, 1975) the i-truncation of AG(d, q) is a subscheme if and only if i = 0, i.e. G(S) is AG(d, q) itself; (iii) (a) (Cameron, 1975) S(t, k, v) is a subscheme if it has at most It/2J + 1 different sizes of pairwise intersections of blocks (hyperplanes); (b) (anonymous referee) S(t, k, v) is a subscheme if it has at most such sizes and if the complement of any block is a block; l(t + 1)/4] + 1 (iv) (a) (Cameron, 1975) ifS(3, k, v) is a subscheme then v < 2 + k(k - 1)(k - 2)/2;
(b) (Atsumi, 1979) if S(t, k, v) is a subscheme then v < k4 All
kl
1k/2).
known S(t, k, v)-subschemes are inversive (Mobius) planes
M. Deza
62
I(q) = S(3, q + 1 = k, q2 + 1 = v) with three classes; S(5, 6, 12) is so because of case (iii) (b) above, and S(3, 4, 8), S(3, 6, 22), S(4, 5, 11) (the unique S(4, 5, v)-subscheme), S(4, 7, 23), S(5, 8, 24), and S(3, 12, 122) (if it exists) are so because of case (iii)(a) of the above theorem.
The triffid with a = (0, 1, 3, 9, 81) (i.e. the smallest one) is not a subscheme but its Mr-1 has a partition into two association schemes (Deza & Hamada, 1980).
2.4. PMDs of Rank 4 The central problem in PMD studies is to find all possible a-sequences, a = (0, 1, (X2, ..., ar). For r = 3 the concepts of PMD (with a = (0, 1, a2, a3)) and BIBD S(2, a2, a3) coincide: so one can use the deep but still incomplete information obtained on BIBDs. Below we survey the known information
on the next, much more difficult and messy, case of rank 4. Throughout section 2.4 we shall consider, a PMD G(S) of rank 4 with a = (0, 1, a2 =1, a3 = k, a4 = v). In sections 2.4 and 2.5 we shall denote 1-, 2- and 3-flats by the terms points, lines, and planes. Denote A = (v -1)/(k - 1). The existence of our PMD G(S) with a = (0, 1, 1, k, v) implies the existence of the four following related BIBDs: (1) D1 = S(2, 1, k), in which points and blocks are points and lines of G(S) contained in a given plane, i.e. D1 is the restriction of G(S) on this plane;
(2) D2 = S 2,
k-1, v-1 1-1 1-1
,
which is the geometrization of the contraction
of G(S) on some point x c- S; (3) D3 = S(2, 1, v), in which points and blocks are points and lines of G(S), i.e. D3 is the 1-truncation of G(S); (4) D4 = S2(2, k, v), in which points and blocks are points and planes of G(S).
(A) The intersection of any two blocks of D4 has size 0, 1, or 1.
The existence of D1, D2, D3, D4 and condition (A) are necessary for the existence of G(S).
Proposition 2.3.1(ii) gives (for the case of rank 4) that the existence of D4 satisfying (A) is sufficient for the existence of G(S), if v > vo for some polynomial vo(T, k) of k.
The necessary conditions of Murty, Young & Edmonds (1970) (of existence of a PMD in terms of divisibility of some numbers related to a) take, for our PMD, the form:
(B) All the num bers are integers.
k-1 k(k-1) v-1 I -l' 1(1 -1)' k - l
(v-1)(v-l) v(v-1)(v-1) 2'
(k - 1)(k - 1)' k(k - 1)(k - 1)
Perfect Matroid Designs
63
The existence of some BIBDs with the parameters of D1, D2, D3, and D4, and the validity of (B), do not together imply the condition (A) on a BIBD with the parameters of D4; an example (given by R. M. Wilson, 1982) is provided by (1, k, v) = (3, 19, 307).
Denote by (b;, v r;, ki, 2;) the parameters of the BIBDs D;, 1 < i < 4. Fischer's (1966) properties of a BIBD give the following necessary condition for the existence of G(S):
(C) b,ki = v,r r;(k; - 1) = 2i(v; - 1), and r, > k; for i = 1, 2, 3, 4. The case r; = k; corresponds to a symmetric BIBD D;, Al = 22 =23 = 1; so if D1, D2, or D3 is symmetric then it is a projective plane. 2.4.1. Proposition. (i) D1 is symmetric if and only if G(S) is PG(d, q): (d - 3), d > 3; (ii) D2 is symmetric if G(S) is PG(3, q), AG(3, q), or S(3, 6, 22); (iii) D3 is not symmetric; (iv) D4 is symmetric if and only if G(S) is PG(3, q). (i) is given in Ray-Chaudhuri & Singhi (1977), (iv) follows from Dembowski
& Wagner (1960). It can be shown that if G(S) has rank at least 5 then D2 is symmetric if and only if G(S) is PG(d, q) or AG(d, q). If the property (A) holds (that is, blocks of D4 have only 0, 1, or 1 as the size of pairwise intersections), Deza & Singhi (1980) gives that: (1) there exists
a point x c S such that a n b 0 {x} for any blocks a, b of D4 (i.e. planes of G(S)) if and only if D2 is symmetric; (2) any an b is non-empty if and only if
(k-1)(kl-k+l)+ll 22-2(2 k1)(1-1)/\k-11=0. Using (C) one can easily calculate the parameters (b;, v ri, ki, 2i) of D.,1 < i < 4.
Now, denote t = (k -1)/(l - 1), u = kt/l and recall that 2 = (v - l)/(k - 1). We shall call the triplet (1, k, v) of integers, 0 < 1 < k < v, admissible if it satisfies
conditions (B) and (C). It is easy to check that (1, k, v) is admissible if and only if 2 <, 1 <, t < 2 and all the following numbers are integers: t, A, t(t - 1)/l, 2(2 - 1)/t, 2(2 - 1)(2t - 2 + 1)/u (it will be equivalent to require that t, A, u, b2, b4 are integers). The extreme cases of the inequalities 2 < 1 < t < A are given below: 2.4.2. Proposition. (i) I = 2 if and only if G(S) is S(3, k, v);
(ii) 1 = t > 2 if and only if G(S) is PG(d, q = t - 1): (d - 3) (recall that G(S): i
denotes truncation);
M. Deza
64
(iii) A = t if and only if D2 is a projective plane of order t - 1 (i.e. we are in case (ii) of 2.4.1);
(iv) (Buekenhout, 1969) 1 = t - 1 > 3 if and only if G(S) is AG(d =logr_
1 v,
q=t- 1): (d-3); (v) (Hall, 1960) 1 = t - 1 = 3 if and only if G(S) is a triffid (see section 2.5) or AG(log3v, 3).
For case (i) of Proposition 2.4.2 we recall that the existence of S(3, k, v) will imply the existence of S(2, k - 1, v - 1). Hanani (1963) gives that S(3, 4, v)
exists if and only if v-2, 4 (mod 6), v >, 8. For any k such that k - 1 is a power of a prime there exists an inversive plane I(k - 1) of order k - 1 that is an S(3, k, (k - 1)2 + 1). In case (ii) of Proposition 2.4.2 the dimension d of PG(d, q) is defined by v = (qd+ 1 - 1)/(q - 1), and PG(d, q) (for our case d > 2) exists if and only if q = t - 1 is a power of a prime. In case (iv) of Proposition 2.4.2 we have v = Id and our G(S) exists if and only if q = t - 1 is a power of a prime. Case (v) is considered in section 2.5. One can see, for example, that `smallest' parameters (1, k, v), such that BIBDs with the parameters of D 1, D2, D3 exist but a PMD is unknown, are (3, 13, 183), (3, 13, 313), (3,15, 183), (3, 15, 675), (3, 19, 307), (3, 19, 1027), (4, 25, 1201), and (4, 25, 8404). In all these cases the existence of a G(S) requires the existence of a D4 = Sz(2, 1, v) with only (0, 1, 3) (for l = 3) and (0, 1, 4) (for l = 4) as sizes of pairwise intersections of blocks.
2.5. Triffids The triffid is a special PMD of rank 4 with a-sequence (0, 1, 3, 9, 3"). The
motivation to study it comes from combinatorics (automorphisms and extensions of an S(2, 3, v)), algebra (groups generated by 3-transpositions, weak associativity, generalization of mean by quasigroups), and even from algebraic geometry (cubic hypersurfaces). In this section we shall first present our objects in their different equivalent forms: special loops, matroids, Steiner triple systems, and also quasigroups and groups. Afterwards, we shall give
(in appropriate terms of loops) some of their parameters, bounds on parameters, and the cases of equality and small parameters. A commutative Moufang loop (CML) is a generalization of an Abelian group in the following direction - the law of associativity is replaced by a weaker property (the commutative version of the Moufang identity): (A quasigroup is a set with a binary operation o such that both left and right translations are bijections; a loop is a quasigroup with an element 1, such that l o x= x o 1= x for all elements x.) Below we consider only the finite exponent 3 CMLs (exp. 3 CMLs), i.e. CMLs in which each element, except 1, has order 3. They are important for
Perfect Matroid Designs
65
us because they provide a class of PMDs. Also, Bruck (1946) gives that for any CML E the central quotient E/Z(E) (see the definition of center, Z(E),
below) is an exp. 3 CML and that any finite CML is a direct product of some Abelian group and some CML in which each element has order a power of 3. Any of our CMLs, i.e. a finite exp. 3 CML, has order 3" for some n. The simplest example of a CML is Z3 X Z3 x ... X Z3 = Z3", an Abelian elementary 3-group where Z3 is the group on {0, 1, 2} with addition modulo
3. Zassenhaus (p. 423 of Bol, 1937) constructed a non-associative CML on the set of all 81 sequences x = (xr, x2, x3, x4) with xi = 0, 1, 2. He defined a product by
x'y=(xr+Yi, x2+Y2, X3 +Y31 x4+Y4+(x3-Y3)(x1Y2-x2Yi)) (all sums above are modulo 3). Denote this CML by E4 and E" the CML Z3_4 x E4 of order 3". We will see below that E. is `close' to Z. A triffid on a given CML E is a PMD of rank 4 such that its points (1-flats) are just elements of E, its lines (2-flats) are all 3-subsets L(x, y) = {x, y, (x y)2 }
for any x, y e E with y 0 1, x, x2, and its planes (or 3-flats or hyperplanes since our PMD has rank 4) are 9-subsets H(x, y, z) =
x,
y,
z,
t,
(z t)2
(x'z)2,
(Y't)2,
(x' t)2
for any x, y, z c- E that are not in a line (here t = [(X. y)2 (X. z)2 ] 2). So, a triffid
is a PMD with a = (0, 1, 3, 9, 3"). Any PMD with such an a comes from
some CML E and in the special case E = Z" this PMD is just an (n - 3)-truncation AG(n, 3): (n - 3) of an affine geometry over GF(3). Any hyperplane (i.e. plane) of a triffid is an affine plane AG(2, 3). The set of all lines L(x, y) of a triffid is an S(2, 3, 3") such that any triangle (i.e. 3 points not in a line) generates an affine plane. We call such an S(2, 3, v) a Hall triple system (HTS). Any HTS is the set of lines of some triffid. The result in Buekenhout (1969) gives that any PMD with a = (0, 1, A, A2, v) such that A > 4 and any triangle generates an affine plane is AG(n, A): (n - 3), so A is a prime power. The triffids, i.e. the case A = 3 (and also S(3, 4, 10) for A = 2, where all planes are AG(2, 2)s) are exceptional with respect to this result.
So, CML and HTS are equivalent concepts. They are also equivalent to each of the following 3 structures:
(1) S(2, 3, 3") such that any symmetry e-" is an automorphism, or, in a x other words, for any point a from our set E of 3" points there exists an involution moving exactly E - {a} (this was proven in Hall, 1960); (2) a symmetric distributive quasigroup (E, o), i.e. a quasigroup such that all
translations are automorphisms and x a y = z is independent of permutations of x, y, z; we say that it has identity e if e o e = e,
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66
e o x = x o e = x (for any x e E) for some e e E; one can get a CML (E,-) from such a quasigroup (E, e, °) by defining x y = (e o x) c (e a y); (3)
a group (G, A), i.e. a group G generated by a subset A such that x2 = 1 = (xy)3 and xyx e A for any x, y e A and such that the commutative
center of G is just {1} (these groups introduced by Fischer, 1966, are considered in Beneteau, 1978); one can get such a group (G, A) from a symmetric distributive quasigroup (E, o) by choosing as generating set A the set of all automorphisms x =R(y) = x o y of this quasigroup. We will now consider the following parameters of the CML E: (1) the order IEI = 3"; (2) the dimension d(E), i.e. the size of any minimal subset generating E; (3) the order IZ(E)I of the center (the associative center Z(E) of the CML E is the Abelian 3-group {z e E I d x, y c- E}); z= x-
(4) the nilpotency class k(E) (the central quotient E/Z(E) is a CML E('), E(')/Z(E(')) is a CML V), etc.; k(E) is the smallest number k such that E('-') is an Abelian 3-group, with E(° = E); (5) the rank3(E) is the 3-rank, i.e. the rank over GF(3), of the incidence matrix (lines-points) of the Steiner triple system S(2, 3, 3") of all lines of E.
2.5.1. Proposition. (i) (Beneteau, 1980a; Sandu, 1979) d(E) < n with equality if and only if E = Z3; (ii) (trivial) IZ(E)I < IEI with equality if and only if E = Z3, Ik(E)I > 1 with equality if and only if E = Z3; (iii) (Doyen, Hubaut & Vandensavel,1978) rank3 (E) ` IEI - n + 1 with equality if and only if E = Z3. It is easy to see that IE41 = 34, Z(E4) = {(0, 0, 0, 0, 0), (0, 0, 0, 1), (0, 0, 0, 2)},
E4 is generated by {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0)}, k(E4) = 2, and so IE" I = 3", IZ(E" )I = IZ(Z3-4) x Z(E4)I = 3" -3 , d(E") = n - 1, k(E") = 2. Hamada (1981) gives rank 3(E") = 3" - n. So E" is very close to an Abelian 3-group; moreover, it is closest to Z3 in the following sense - for any non-associative CML E with IEI = 3", Beneteau (1980a) gives IZ(E)I S 3n-3 with equality if
and only if E = E. The order IEI of a non-associative CML E is 3" for some n >, 4; one exists (for example, E") for any such n. 2.5.2. Proposition.
(i) (Hall, 1960) E4 is the unique non-associative CML of order 34; (ii) (Beneteau, 1980a; Smith, 1982) E5 is the unique non-associative CML of order 35; (iii) (Beneteau,1980b) there exist exactly three non-associative CMLs of order
Perfect Matroid Designs
67
36: E6, I5, and 14, ofdimensions 5, 5, and 4 (see also Kepka & Nemec,1981;
Roth & Ray-Chaudhuri, 1984).
For the dimension d(E) of any non-associative CML E we have 3 1< d(E) 5 log3IEI - 1, d(En) = n - 1, and d(E) > 4 for E :A E4.
2.5.3. Proposition. (Beneteau, 1980b) Let d(E) = 4; then IEI < 312 and for IEI < 38 there exist exactly eight CMLs with d(E) = 4; for each of the orders 34, 35, 36, 37, 312 there exists exactly one such CML (Z3, E5, I41 J4, L4). There exists for any m the free CML of dimension m, denoted Lm, such that any CML E with d(E) = m is a representation (homeomorphic image) of Lm. So IEI 5 ILd(E)I and k(E) 5 k(Ld(E)); L3 = E4. In Manin (1974) the problem arises of determining ILm I; it is known only that ILm I = 34, 312, 349, 3220, 3102s 34592 for m = 3, 4, 5, 6, 7, 8 (Beneteau, 1980b; Smith, 1982). So, for example, the bound IEI <, 312 for CMLs with d(E) = 4, given in Proposition 2.5.3, follows from IEI 5 ILd(E) I and IL41 = 312
The dimension, d(E), is also linked with the order of the Frattini subloop t(E) of CML E by the equality l b(E)I = IEI/3d(E) This subloop D(E) is defined
as the intersection of all maximal subloops of E; also, I(E) is the set of all
non-generators, i.e. elements that can be taken away from any subset generating E. Further, we remark that any CML is diassociative, i.e. any two elements generate an Abelian subgroup.
IZ(E)l is a power of 3, because Z(E) is an Abelian 3-group. For any non-associative CML E with IEI = 3" we have 1 < 1og3IZ(E)I < n - 3.
The left bound is given in Bruck (1958), the right bound in Beneteau (1980a). Roth & Ray-Chandhuri (1984) gives that each of numbers 1, 2, 3, ..., n - 3 is log3IZ(E)I for some CML E with IEI = 3" (of course, we have to take n:5). IZ(E")I =
3"-3 and IZ(Lm)I = 3(m/3)
We now consider the central nilpotency class k(E) of the CML E. k(Z3) = 1 and k(E") = 2. Beneteau (1978) gives 34 < IEI < 3d+(dl3) for E with k(E) = d and 38 < IEI < 3d+(d/3)+4(d+ 1/5) for E with k(E) = 3, d(E) = d (all four bounds are exact.) All CMLs E with k(E) = 2 are classified in Bruck (1946) and also in Kepka & Nemec (1981) and in Roth & Ray-Chaudhuri (1984). 2.5.4. Proposition.
(i) (Bruck, 1958) k(E) 5 d(E) - 1 (Bruck-Slaby); (ii) (Beneteau, 1978; Malbos, 1978; Smith, 1978): k(Lm) = m - 1 if m >, 2 (Beneteau-Malbos- Smith). Here we describe the construction given in Malbos (1978) for k(Lm) = m - 1.
M. Deza
68
Let V be a vector space over GF(3) and A * V = ® A2i+1 V the submodule =o
of odd sums A2i+1 Vof an exterior algebra A V of V Denote EM = (A * V) x (A * V)
and define a product ° on E. by -xiYi(x2-Y2),x2+Y2-(x1Y2+x2y0(x2-y2))
(x1, x2)°(Yi, Y2)=(xl+Y1 (EM,
is
a CML and for
{x ..., xm, Yi,
any linearly independent 2m-subset
., Ym} of V(m > 1) the m-subset {(x1, y1)I i = 1, 2, ..., m} of EM
generates a subloop E of (EM, °) of class m - 1. So m - 1 = k(E) = d(E) - 1 and k(E) < k(Lm) <, m -1 implies k(Lm) = m - 1. There exists for any m and for any k, k S m - 1, the free CML of dimension m and nilpotency class k; denote it Lm,k. Then I EI S ILd(E),k(E) I and also Lm,m _ i = Lm. Now Beneteau & Lacaze (1980) gives n )k-2
\/
1)
log3ILm,kl <m+ (3) +4(m
5
/J
-1
2
(2n)
-1
moreover ILm,2l = 3m+(m/3) Also in Beneteau & Lacaze (1980) it is proved
that the order of the automorphism group IAut El of any 3-CML E of m
dimension m divides the number (IEIm/3(m+1/2))
H(3'- 1) and that in the
i=i
special case E = Lm,k the order is equal to that number. There is a developed theory of infinite CMLs. An example of an infinite
CML is the loop of octonions. Infinite CMLs have connections with differential geometry (3-webs), topological algebra (CMLs on topological spaces, Lie loops), and algebraic geometry (the cubic hypersurfaces of Manin).
Basic work on CMLs was carried out by Bruck (1946, 1958). HTSs and links with groups (G, A) were given by Hall (1960). Young (1973) observed
the link between CMLs and HTSs, and PMDs. Symmetric distributive quasigroups (equivalent to CMLs) were studied by Beloussov and his students (see Sandu, 1979, and some other papers in the same book) and (in terms of symmetric medial spaces) by Soublin (1971) and, recently, Malbos (1978). A
promising (but not yet completely developed) link of CMLs with cubic hypersurfaces, and many other developments, were given by Manin (1974). In recent years CMLs have been studied intensively by Beneteau (for example, Beneteau, 1978, 1980a, b; Beneteau & Lacaze, 1980), Smith (for example, 1978, 1982), Kepka and Nemec (for example, Kepka, 1978; Kepka & Nemec, 1981), and Roth & Ray-Chaudhuri (1984); some new observations on CMLs are given in Deza & Hamada (1980), Deza & Singhi (1980), and Hamada (1981).
2.6. Some Close Relatives of PMDs and Further Directions An equicardinal matroid M(S) is a matroid such that all its hyperplanes have
Perfect Matroid Designs
69
the same cardinality k. It was introduced in Murty (1970) and, in fact, the PMD was introduced later just as a special case of equicardinal matroid. Kestenband & Young (1978) gives information on such matroids and, especially, on the case when ISI - k is a prime power p". These equicardinal matroids (which are a direct generalization of PG(d, q) having ISO - k = q°) are shown in Kestenband & Young (1978) to be `almost' PMDs; they are
classified for the cases n = 1, 2 and for the case when M(S) has rank 3. A matroid design (MD, for short) is a matroid whose hyperplanes form a BIBD, i.e. a (b, v, r, k, A) configuration on 1-flats. Any PMD is an MD. Any BIBD with A = 1 is an MD (moreover, a PMD) of rank 3; it is easy to see that any non-trivial BIBD with A = 2 is never an MD. Theorem 3 in section 12.3 of Welsh (1976) gives the necessary conditions for a BIBD with 2 > 2
to be an MD. Kantor (1969) showed that a symmetric BIBD, i.e. a (v, k, .1)-configuration, is an MD if and only if either 2 = 1, or (q - 1) (v, k, 2) = (q"+ t - 1, q" - 1, q"-' - 1) for some prime power q and the BIBD is the point-hyperplane design of PG(n, q). Section 12.3 of Welsh (1976) also contains information on base designs, i.e. matroids whose circuits are the blocks of a BIBD. Also, finding circuit designs, i.e. matroids whose bases are the blocks of a BIBD, is, of course the dual problem to finding MDs. Another close relative of PMDs of rank 4 is a planar M-space (Buekenhout & Deherder, 1971). A planar space (or respectively a linear space) is exactly a combinatorial geometry of rank 4 (or respectively 3). A planar M-space is a planar space such that any of its planes is isomorphic to the given linear space M. Any PG(d, q): (d - 3), AG(d, q): (d - 3), or triffid is a planar M-space.
An example of a non-trivial (i.e. with at least two planes) planar M-space with two different sizes of lines is known, but only one; Buekenhout & Deherder (1971) show that the number of such planar M-spaces is finite for any given M. Brylawski (1979) considers uniform matroids, i.e. matroids such that all upper intervals (in the lattice L of all flats) of the same rank have the same v) = l4'j ] (u, v) for any flats Whitney numbers of the second kind: x, y of the same rank and flats u, v with v s u. Any PMD is a uniform matroid. Theorems 4.5 and 4.7 of Brylawski (1979) give a criterion for a matroid to
be a PMD in terms of rank invariants and calculate the characteristic polynomial of a PMD as a function of its a-sequence. Among other types of uniformity, considered in Brylawski (1979), we mention latticially homogeneous
matroids, i.e. [0, x] = [0, y] for all bottom intervals (of the lattice L of all flats) with x, y of the same rank (these are a special case of PMD), and latticially uniform matroids, i.e. [x, S] = [y, S] for all upper intervals of L with x, y of the same rank (these are a special case of uniform matroids and not necessarily PMDs). Another generalization of the lattice of a PMD is
M. Deza
70
considered in Neumaier (1981). PMDs also appear in the following lattice-theoretic context (Theorem 9.1 of Doubilet, Rota & Stanley, 1972). For any upper semimodular lattice L of triangular type (i.e. the number of all maximal chains in any segment [x, y]
of L depends only on the ranks of x and y) there are PMDs L1, L2, ..., L, and an upper semimodular atomic lattice L,+, of triangular type, such that L is isomorphic to the lattice obtained by identifying the top of Li with the bottom of Li +, for 1 < i < r. The papers by Deza (1978) and by Deza, Erdos & Frankl (1978), in which was given Proposition 2.3.1 (a characterization of all PMDs with large ar as the case of equality in an upper bound for JA(T, k, v = a,)I; here T is a set of integers) contain also several generalizations of the inequality such that the equality will correspond to a generalization of a PMD. Denote by AZ(T, k, v) any family of k-subsets of a given v-set S such that any (A + 1)-wise intersection
of them is a number from the set T= {lo, 11, ..., 'r-21 of integers, 0 < to < 11 < ... < 1, _ 2. Denote by Mz(T, k, v) a family of k-subsets of S, called `hyperplanes', such that for any (r-2)-flat (of given PMD of rank r- 1 with a = (la, 11, ..., 'r-21 v)) and point outside it there are exactly A `hyperplanes' containing this (r - 2)-flat and point. Any S.(t, k, v) is an M,({0, 1, ..., t - 11, k, v); also we obtain M2(T, k, v), with A= i
fl (v -1(`))/(k - l(`)), from a PMD with a = (la, 11, ..., lr_2, l(l), _1
l°, k, v), lr_2 < 1
> < ...
10) < k, by deleting all flats of sizes
1(l),
1(2),
...,
..., 10.
Deza (1978) gives, for v > v0(2, T, k), that I AA(T, k, v)l < 2 fi (v - l)/(k -1) with IcT
equality if and only if AZ(T, k, v) is an Mz(T, k, v); also that any MA(T, k, v) with v > vo is complete, i.e. any number from T is the size of some (A + 1)-wise intersection of sets from it. Let A be an A(T, k, v), T = {lo, 1, , ..., lr _ 2 }, 0 S to < ... < 1, _ 21 such that
A is a subset of the hyperplane family of some PMD G'(S) with 'X=(10, it, ..., 4-211"), ..., l('), k, v). Then lAl ,< H (v - l)/(k - 1)) (in contrast lcT
with Proposition 2.3.1 we have no condition v > vo); the case of equality is a special PMD M(S) which can be seen as an S(t = r - 1, k, v) in the PMD G'(S). In the special case when the hyperplane family of G'(S) is partitioned by such A(T, k, v)s we will call G'(S) i-resolvable (with i = r - 1). A 1-resolvable PMD is just an affine resolvable PMD (M{lo }, k, v) if to = 0 forms a partition
of S). In design theory some cases of i-resolvability of S(k, k, v) are also considered. Another useful concept in relation to the resolvability of a PMD is affine resolvability of the configuration dual to the BIBD of hyperplanes and points (Deza, 1978).
Finally, in Deza & Frankl (1984) a new notion, squashed design, is
Perfect Matroid Designs
71
introduced. Squashed designs are more general than PMDs; they also include some geometric structures on the set of permutations. Deza & Frankl (1984),
extending Proposition 2.3.1, characterized squashed designs as families A(T, k, v) of maximum size and belonging to some meet semi-lattice (v > vo (T, k))
F of subsets of S. If F is the lattice of flats of some PMD, then it is a design within this PMD as above.
References Atsumi, T. (1979). An extension of Cameron's result on block schematic Steiner systems, J. Comb. Theory Ser. A 27, 388-91. Beneteau, L. (1978). Problemes de majorations dans les quasigroupes distributifs et les groupes de Fischer, in Acres Colloque `Algebra Appliquee et Combinatoire', pp. 22-34. Univ. Sci. et Med. de Grenoble. Benteau, L. (1980a). Topics about Moufang loops and Hall triple systems, Simon Stevin 54, 2, 107-24.
Beneteau, L. (1980b). Une classe particuliere de matroides parfaits, in Actes Colloque FranceCanada de Combinatoire (Montreal, 1979), North-Holland, Amsterdam. Discrete Math. 8, pp. 229-32. Beneteau, L. & Lacaze, J. (1980). Groups d'automorphismes des boucles de Moufang commutatives, Europ. J. Comb., 1-4, 299-309. Bol, G. (1937). Gewebe and Gruppen, Math. Ann. 114, 414-31 (especially p. 423). Bose, R. C. (1963). Strongly regular graphs, partial geometries and partially balanced designs, Pacific J. Math. 13, 389-419. Brini, A. (1980). A class of rank-invariants for perfect matroid designs, Europ. J. Comb. 1-1, 33-8. Bruck, R. H. (1946). Contributions to the theory of loops, Trans. Amer. Math. Soc. 60, 245-354. Bruck, R. H. (1958). A Survey of Binary Systems. Springer, Berlin, Gottingen, and Heidelberg. Brylawski, T. H. (1979). Intersection theory for embedding of matroids into uniform geometries, Stud. Appl. Math. 61, 211-44. Brylawski, T. H. (1986). Constructions, in N. White (ed.), Matroid Theory, Cambridge University Press. Buekenhout, F. (1960). Characterization des espaces affine basee sur la notion de droit, Math. Z. 111, 367-71. Buekenhout, F. & Deherder, R. (1971). Espaces lineaires finis a plans isomorphes, Bull. Soc. Math. Belg. 23, 348-59. Cameron, P. J. (1975). Two remarks on Steiner systems, Geom. Dedicata 4, 403-18. Cameron, P. J. (1980). Extremal results and configuration theorems for Steiner systems, in C. Lindner & A. Rosa (eds), Topics on Steiner systems, Ann. Discrete Math. 7, pp. 43-63. NorthHolland, Amsterdam. Dembowski, P. & Wagner, A. (1960). Some characterizations of finite projective spaces, Arch. Math. 11, 465-69. Deza, M. (1978). Pavage generalise parfait comme generalisation de matroide-configurations et de simples t-configurations, in Problemes Combinatoires, Coll. Int. CNRS, No. 260, pp. 97-100. Paris-Orsay. Deza, M. & Frankl, P. (1984). Injection Geometries, J. Comb. Theory Ser. B 36, 31-70. Deza, M. & Hamada, N. (1980). The geometric structure of a matroid design derived from some commutative Moufang loop and a new MDPB association scheme, Technical Report No. 18, Statistical Research Group, Hiroshima University. Deza, M. & Singhi, N. M. (1980). Some properties of perfect matroid designs, in Proceedings of the Symposium on Combinatorial Mathematics and Optimal Design, Ann. Discrete Math. 6, pp. 57-76. North-Holland, Amsterdam. Deza, M., Erdos, P. & Frankl, P. (1978). Intersection properties of systems of finite sets, Proc.
72
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London Math. Soc. (3) 36, 369-84. Deza, M., Frankl, P. & Hirschfeld, J. W. P. (1985). Sections of varieties over finite fields as large intersection families, Proc. London Math. Soc. (3) 50, 405-25. Doubilet, P., Rota, G. C. & Stanley, R. (1972). On the foundation of combinatorial theory (VI): The idea of generating functions, in Proceedings of thje Sixth Berkeley Symposium of Mathematical Statistics and Probability Vol. 2, pp. 267-318. University of California Press, Berkeley. Doyen, J., Hubaut, X. & Vandensavel, M. (1978). Ranks of incidence matrices of a Steiner triple system, Math. Z. 163, 251-59. Doyen, J. & Hubaut, X. (1971). Finite regular locally projective spaces, Math. Z. 119, 83-8. Fischer, B. (1966). Finite groups generated by 3-transformations, Invent. Math. 13, 232-76. Frankl, P. (1983). Personal communication. Gross, B. H. (1974). Intersection triangles and block intersection numbers of Steiner systems, Math. Z. 139, 87-104. Hall, M. (1960). Automorphisms of Steiner triple systems, IBM J. Res. Dev. 4, 464-72. Hamada, N. (1981). The geometric structure and the p-rank of an affine triple system, derived from a nonassociative Moufang loop with the maximum associative center, J. Comb. Theory Ser A 30, 285-97. Hamada, N. & Tamari, F. (1975). Duals of balanced incomplete block designs derived from an affine geometry, Ann. Stat. 3, 926-38. Hanani, H. (1963). On some tactical configurations, Can. J. Math. 15, 702-22. Kantor, W. M. (1969). Characterizations of finite projective and affine spaces, Can. J. Math. 21, 64-75. Kepka, T. (1978). Distributive Steiner quasigroups of order 35, Comment. Math. Carolin. 19-2, 389-401. Kepka, T. & Nemec, P. (1981). Commutative Moufang loops and distributive groupoids of small order, Czechoslovak Math. J. 31 (106), 633-69. Kestenband, B. C. & Young, H. P. (1978). J. Comb. Theory Ser. A 24, 211-34. Malbos, J.-P. (1978). Sur la classe de nilpotence des boucles commutatives de Moufang et des espaces mediaux. C.R. Acad. Sc. Paris Ser. A 287, 691-93. Manin, J. I. (trans. M. Hazewinkel) (1974). Cubic Forms, North-Holland, Amsterdam. Murty, U. S. R. (1970). Equicardinal matroids and finite geometries, in Proceedings of the International Conference on Combinatorial Structures and Applications, pp. 289-91. Gordon & Breach, New York. Murty, U. S. R., Young, H. P. & Edmonds, J. (1970). Equicardinal matroids and matroid-designs, in Proceedings of the Second Chapel Hill Conference on Combinatorial Mathematics and Applications, pp. 498-547. Chapel Hill, North Carolina. Neumaier, A. (1981). Distance matrices and n-dimensional designs, Europ. J. Comb. 2-2, 165-72. Ray-Chaudhuri, D. K. & Singhi, N. M. (1977). A characterization of line hyperplane design of projective space and some extremal theorems for matroid-designs, in H. J. Zassenhaus (ed.), Number Theory and Algebra, pp. 289-301. Academic Press, New York. Roth, R. & Ray-Chaudhuri, D. K. (1984). Hall triple systems and commutative Moufang exponent 3 loops: The case of nilpotence class 2, J. Comb. Theory Ser. A 36, 129-62. Sandu, N. I. (1979). On centrally nilpotent commutative Moufang loops, in Quasigroups and Loops (in Russian), pp. 145-55. Stiintza, Kishiniev. Smith, J. D. H. (1978). On the nilpotence class of commutative Moufang loops, Math. Proc. Cam. Phil. Soc. 84, 387-404. Smith, J. D. H. (1982). Commutative Moufang loops and Bessel functions, Invent. Math. 67, 173-86. Soublin, J: P. (1971). Etude algebraique de la notion de moyenne, J. Math. Pures et Appl. Ser. 9 50, 53-264. Young, H. P. (1973). Affine triple systems and matroid designs, Math. Z. 132, 343-59; Math. Rev. 50, 142. Welsh, D. J. A. (1976). Matroid Theory, Academic Press, London and New York. White, N. (ed.) (1987). Combinatorial Geometries, Cambridge University Press. Wilson, R. M. (1982). Personal communication.
3
Infinite Matroids JAMES OXLEY
The many different axiom systems for finite matroids given in Chapter 2 of White (1986) offer numerous possibilities when one is attempting to generalize
the theory to structures over infinite sets. Some axiom systems that are equivalent when one has a finite ground set are no longer so when an infinite ground set is allowed. For this reason, there is no single class of structures
that one calls infinite matroids. Rather, various authors with differing motivations have studied a variety of classes of matroid-like structures on infinite sets. Several of these classes differ quite markedly in the properties possessed by their members and, in some cases, the precise relationship between particular classes is still not known. The purpose of this chapter is to discuss the main lines taken by research into infinite matroids and to indicate the links between several of the more frequently studied classes of infinite matroids. There have been three main approaches to the study of infinite matroids, each of these being closely related to a particular definition of finite matroids. This chapter will discuss primarily the independent-set approach. Some details of the closure-operator approach will also be needed, but a far more complete treatment of this has been given by Klee (1971) and by Higgs (1969a, b, c). The third approach, via lattices, will not be considered here. This approach is taken by Maeda & Maeda (1970) and they develop it in considerable detail. Throughout this chapter, the Axiom of Choice will be assumed.
3.1. Pre-independence Spaces and Independence Spaces The first class of infinite matroids that we consider is obtained essentially by deleting the references to finite sets in the independence axioms (il)-(i3) of Chapter 2 of White (1986). We note, however, that the finiteness of I1 and I2 in (i3), which is implicit when S is finite, is made explicit when ISI is unrestricted.
James Oxley
74
A pre-independence space Me(S) is a set S together with a collection J of subsets of S (called independent sets) such that
(il) J :A 0. (i2) A subset of an independent set is independent. (i3') (Finite augmentation) If I1 and I2 are finite members off with 112 I > II1 I1
then there exists x in I2 - I1 such that I1 u x e J. Generalizing the terminology of finite matroid theory, we call a subset X of S dependent if X 0J. A circuit of Me(S) is a minimal dependent set, and a basis of Me(S) is a maximal independent set. The notation Y cc X indicates that Y is a finite subset of X. Although pre-independence spaces seem to be natural objects for study, they have received little attention in their own right primarily because they fail to possess many of the fundamental properties of finite matroids.
3.1.1. Example. Let S = l8, the set of real numbers, and S be the set of all countable subsets of S. Then S is the collection of independent sets of a pre-independence space Me(S). However, Mr(S) has no circuits and no bases.
In the face of such examples, it is natural to strengthen one's axiom system. As with finite matroids, a principal example of a pre-independence space is obtained from a vector space V In this case, we let S be an arbitrary subset of V, and S be the collection of subsets of S that are linearly independent in V Such a pre-independence space satisfies the following additional condition. (14) (Finite character) If X c S and every finite subset of X is in S, then X is in J.
Much of the work done on infinite matroids has been algebraically motivated and for this reason (il), (i2), and (i3') have frequently been augmented by (14). We shall call a pre-independence space satisfying (14) an independence space. Such structures are also commonly referred to asfinitary matroids (Bean, 1976; Higgs, 1969a; Klee, 1971). In this section we examine
the properties of independence spaces. In the next section we shall add different conditions to (il), (i2), and (i3') with different consequences.
3.1.2. Example. It is easy to extend the relevant arguments from Chapter 4 of White (1987) to show that the set of partial transversals of an arbitrary family of subsets of a set S is the set of independent sets of a pre-independence space on S. However, this pre-independence space need not be an independence space. For instance, if .1= (X1, X2, X3, ...), and X. = 11, i + 11 for all i, then
every finite subset of Z+ is a partial transversal of t, yet 71+ itself is not. Two immediate consequences of (14) are that every dependent set in an independence space contains a circuit and that this circuit is finite. Thus an
Infinite Matroids
75
independence space is uniquely determined by its collection of circuits. We leave it to the reader to show that the independence axioms, (il), (i2), (i3'), and (14), are cryptomorphic to each of the circuit axioms, the strong circuit axioms, and the basis axioms stated below. 3.1.3. Circuit axioms for independence spaces. An independence space M(S) is a set S together with a collection ' of subsets (called circuits) such that W satisfies (clHc3) (Chapter 2 of White, 1986) together with (C4) Every circuit is finite.
3.1.4. Strong circuit axioms for independence spaces. These are the same as 3.1.3, except that (c3) is replaced by (c3.1) of Chapter 2 of White (1986).
The bases of an independence space also behave similarly to their counterparts in a finite matroid. Indeed, an easy consequence of Zorn's Lemma is that every independent subset of an independence space is contained
in a basis. Hence every independence space is determined by its collection of bases. To obtain a definition of an independence space in terms of bases, one augments the basis axioms (bl)-(b3) of Chapter 2 of White (1986) by the following form of the finite character condition.
(B4) If X is not contained in a basis, then some finite subset of X is not contained in a basis. It is not difficult to extend the axioms (cll){c14) of White (1986) to give a closure-operator definition for independence spaces. We leave the reader to check the details of this (see Exercise 3.1).
3.1.5. Example. Let S be the set of edges of a graph F and let '(F) be the collection of edge-sets of cycles of IF. It was noted in Chapters 1 and 6 of White (1986) that, when IF is finite, '(F) is the set of circuits of a (finite) matroid on S. Since every cycle in an infinite graph is finite, we can extend this immediately to get that '(I) is the set of circuits of an independence space Mr(S) regardless of whether F is finite or infinite. If M(S) is an independence space having 5 as its collection of independent sets, then for X 9 S, let f I X be defined, as for finite matroids, by
5I X = {Yg X: YeJ}.
(3.1)
Clearly 5 I X is the collection of independent sets of an independence space M(X) on X. We call M(X) the restriction of M(S) to X. One can also define the operation of contraction for independence spaces in the same way as for
finite matroids. However, in order to establish that this operation is well defined and that it gives an independence space, we shall require three lemmas.
76
James Oxley
3.1.6. Lemma. Let B1 and B2 be bases of an independence space M(S) and suppose that x E B1 - B2. Then there is an element y of B2 - B1 such that (B 1- x) u y is a basis of M(S).
Proof. This result is an easy consequence of the proof of Proposition 2.1.1 of White (1986).
3.1.7. Lemma. Suppose that M(S) is an independence space and Y X c S. If B1 and B2 are bases of M(S - X), then B1 u Ye J if and only if B2 U YE'f. Proof. Suppose B1 u Ye J, but B2 U YO J. Consider M((S - X) u Y). This is an independence space having B1 u Y as a basis. Moreover, M((S - X) u Y) has a basis B such that B2 9 B c B2 U Y Now B = B2 U Y' where Y' Y. Choose x in Y - Y'. Then x c- (B 1 u Y) - B and so, by Lemma 3.1.6, there is
an element y of B - (B1 u Y) such that ((B1 u Y) - X) u y is a basis of M((S-X)uY).AsY's Y,, we have that yeB2-B1 and soB1 B1uyg; S - X, and B1 uy is independent in M(S - X). This contradicts the fact B1 is a basis of M(S - X) and thereby completes the proof of the lemma. We require one further lemma before defining contraction for independence spaces. If S and I are sets and . ' = (Xi: i e I) is a family of subsets of S, then a choice function for X is a mapping 0: I-*S such that Qb(i)EX1 for all i in I. If J c I, then 0 I, denotes the mapping from J into S defined by ¢ I,(j) = 0(j)
for all jinJ. 3.1.8. Lemma. (Rado's selection principle) Let (X i s i d ) be a family of finite subsets of a set S. For each finite subset J of I, let ¢. be a choice function for (Xi: i e J). Then there is a choice function 0 for (Xi: i e I) such that if J cc 1,
then there is a set K for which J c K cc I and 4I J = 4)K I, Mirsky's book (1971) contains several applications of this result together with a short proof of it using Tychonoff's theorem. We shall not reproduce these here. 3.1.9. Proposition. Suppose that M(S) is an independence space, X s S, and B is a basis of M(S - X). Let
5.X = {YsX: YuBEJ}.
(3.2)
Then 5.X is the set of independent sets of an independence space M.X, the contraction of M(S) to X.
Proof. By Lemma 3.1.7, 5.X does not depend on the basis B chosen for M(S - X). It is clear that 5.X satisfies (i 1) and (i2). Moreover, if Y g X and every finite subset of Y is in 5.X, then every finite subset of Yu B is in J. Hence Yu B E S, and so YES.X. Thus, 5.X satisfies (14).
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To show that ..X satisfies (i3'), we shall use Rado's selection principle. Suppose that U and T are finite members of f.X and I TI > I UI. Then B u T and B u U are in 5. Moreover, for every finite subset Bo of B, we have that M(Bo u To U) is a finite matroid, and so, applying (i3) to Bo u T and Bo U U,
there is an element t of T - U such that Bo u U u t e J. Thus the set SBo = {t e T - U: Bo u U u t e f} is finite and non-empty. Let I be the set of finite subsets of B and suppose that J = {Bt, B21 ..., Bn} cc I. Let B' = U B, i=t
and choose an element t' from SB.. We define a choice function cJ for (SB,, SB2, ..., SBn) by (kJ(Bi) = t' for all i in {1, 2, ..., n}. Let 0 be a choice function for (SBo Bo a I) satisfying the conclusion of Lemma 3.1.8. We show next that 0 maps every element of I to the same element to of T - U. It will follow from this and (14) that B u U u to e S, and hence that U u to e .f.X, as required. To show that the image of 0 is a single element of T - U, suppose
that Bo, Bo a 1, and let J = {Bo, Bo }. Then there is a set K such that
J K cc I and Y'K IJ = 4)IJ. But 4,K(Bo) = ¢K(B') and so ¢(Bo) = 4,(Bo). El It is well known that all bases of a vector space are equicardinal. This result was extended to independence spaces by Rado (1949).
3.1.10. Proposition. If Bt and B2 are bases of an independence space M(S), then IBt I = IB2I.
The proof of this result uses the following infinite extension of Rado's theorem on independent transversals (see Chapter 4 of White, 1987). Note that, since every restriction of an independence space M(S) to a finite set X is a finite matroid, one can define the rank r(X) of X to be the rank of the matroid M(X). 3.1.11. Proposition. Let M(S) be an independence space and . ' = (Xi: i e I) be a family of finite subsets of S. The following statements are equivalent. (i)
r(U X) > IJI for every J cc I. JEJ
(ii) Every finite subfamily of . ' has an independent transversal. (iii) X has an independent transversal.
Proof. The fact that (i) implies (ii) follows from the finite case of Rado's theorem. Moreover, it is clear that (iii) implies (i). We shall complete the proof of Proposition 3.1.11 by using Rado's selection principle to show that (ii) implies (iii).
Suppose L cc I. Then, by (ii), (Xi: i e L) has an independent transversal. Thus there is an injective choice function ¢L for (Xi: i e L) such that 4,L(L) eJ.
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Let 0 be a choice function for (X,: i e I) satisfying the conclusion of Lemma 3.1.8. Then if i1, i2 e I and J' = {i1, i2 }, there is a finite subset K' of I such that Y g K' and 4>K' IJ' _ 4 I , As
4,K. (i2 ), it follows that 4>(i1) s (k('2),
so 0 is an injection and hence 0(1) is a transversal of X. To see that 0(1) e.1, suppose that Y cc 0(1). Then Y = 4>(J) for some J cc I. Now I has a finite subset K such that K 2 J and OK IJ = W IJ Since 4>K(K) e f, we have that Y= 4>(J) = 4>K(J) e J. Thus every finite subset of 0(1) is in J, and so 0(1) e J, as required.
Proof of Proposition 3.1.10. The required result will follow if we can show
that for U, T e f and I UI
operator of the finite matroid N = MI T' u
U Xt (cT'
J/
Then, for all t in
T' - U, the set C, is a circuit of N and so T' c g( U Xt).
Hence
re T'
r(
U Xt)
r(T') =I T'I. Thus, by Proposition 3.1.11, .i' has a transversal;
t e T'
that is, there is an injection from T into a subset of U. Hence I TI < I UI; a contradiction. This completes the proof of Proposition 3.1.10. It follows from Proposition 3.1.10 that in an independence space M(S) one
can define the rank r(X) of an arbitrary subset X of S to be the common cardinality of all bases of M(X). The preceding discussion has shown that a large number of fundamental properties of finite matroids are shared by independence spaces. In particular,
the basic operations of restriction and contraction can be defined for independence spaces. Another important and powerful tool for finite matroids
that one would naturally wish to extend to independence spaces is the operation of orthogonality. We shall show, however, that this cannot be done. Indeed, the lack of a satisfactory theory of orthogonality for independence
spaces has been an important motivating factor in the study of other classes of infinite matroids. Suppose that M(S) is an independence space and let 5* be defined as follows.
5* = {X: S - X contains a basis of M(S)}.
(3.3)
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We leave the reader to check that 5* is the set of independent sets of a pre-independence space M*(S). Evidently, every independent set of M*(S) is contained in a basis of M*(S). The term `cofinitary matroid' has often been used to refer to such pre-independence spaces M*(S) (Bean, 1976; Klee, 1971).
When S is finite 5*, of course, is the collection of independent sets of the matroid orthogonal to M(S). However, as the next example shows, when S is infinite, M*(S) may fail to satisfy the finite character condition.
3.1.12. Example. Let S be an infinite set and k be a positive integer. If Jk = {X c S: IX <, k}, then 5k is the set of independent sets of an independence
space M"(S). However, although every finite subset of S is in 5,*, the set S itself is not. This example plays a central role in the proof of the next theorem. Let S be an arbitrary infinite set and . be the set of independence spaces on S. An orthogonality function A on Y is a mapping from . into . such that for all 5 in 9 we have that A is an involution, that is,
A(A5) = 5
(3.4)
and
(A.f) I X = (5 X )*
for all X (-_(-_S.
(3.5)
The second of these conditions expresses agreement between A and the usual orthogonality for finite methods.
3.1.13. Theorem. There is no orthogonality function on the collection . of independence spaces on an infinite set S.
Proof. Assume that there is an orthogonality function A on Y. Now if k is a positive integer and X cc S, then S - X is infinite, so `Ak I (S - X) contains a basis of M"(S). Thus fk.X = {Q }, hence X E (.fk.X)* and so, by (3.5), X E AS,.
It follows that ASk contains all finite subsets of S and hence A.Sk = 2s Therefore,
if j and k are distinct positive integers, then, by (3.4),
Jj = A(A.%) = A(Aik) = fk; a contradiction. This completes the proof of Theorem 3.1.13. Rado (1966) raised the problem of developing a non-trivial theory of infinite
matroids in which the finite character condition does not feature. The motivation for discarding this condition is increased by Example 3.1.12, which
suggests that the problem with attempting to define orthogonality for independence spaces may arise because the class of independence spaces is too restricted. The rest of this chapter will be concerned with solving Rado's problem.
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3.2. B-matroids In this section we discuss the properties of basis-matroids (B-matroids), a class of pre-independence spaces introduced by Higgs (1969a) whose members
have many of the properties of finite matroids. It is shown that this class contains the class of independence spaces and is closed under restriction and contraction. Moreover, unlike the class of independence spaces, the class of B-matroids is closed under the natural orthogonality function. If .2 is a collection of subsets of a set S, then an 1-subset of S is a subset of S that is a member of Y. A B-matroid MB(S) is a set S together with a collection 5 of subsets of S such that S satisfies (il) and (i2) together with If Ts: X c Sand Te 5, then there is a maximal f-subset of X containing T (IB2) For all X g S, if B1 and B2 are maximal f-subsets of X and x e B1- B21 then there is an element y of B2 - B 1 such that (B 1 - x) u y is a maximal (IB 1)
f-subset of X. It is easy to check that (1B2) implies (i3) and hence that every B-matroid is a pre-independence space. Thus, if MB(S) is a B-matroid, the members of S are called independent sets, and a maximal member of f is called a basis. An immediate consequence of Lemma 3.1.6 is that every independence space is a B-matroid. But, as the following example shows, not every B-matroid is an independence space. 3.2.1. Example. Let S be the edge set of the infinite graph F shown in Figure 3.1 and let S consist of those subsets of S that do not contain the edge set of any cycle or two-way infinite path in IF. It is straightforward to check that S satisfies (il), (i2), (IB1), and (IB2). Hence S is the collection of independent sets of a B-matroid MB(S) on S. However, S does not satisfy (14) because
every finite subset of a two-way infinite path is in 5, yet the path itself is not. Therefore, MB(S) is not an independence space. Figure 3.1.
This example is one member of a class of B-matroids introduced by Higgs (1969c) (see Exercise 3.19).
If MB(S) is a B-matroid having S as its collection of independent sets, then for X c S, let f I X be defined as in (3.1). It is easy to check that S I X is the set of independent sets of a B-matroid, MB(X), the restriction of MB(S) to X.
We show next that the operation of contraction can be defined for
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B-matroids as in (3.2). First note that the argument that was used to prove Lemma 3.1.7 can be applied in this case to give that 5.X does not depend on the basis chosen for f I (S - X). 3.2.2. Proposition. If MB(S) is a B-matroid and X c S, then 5.X is the set of independent sets of a B-matroid MB.X on X. Proof. Let B be a basis of MB(S - X). Clearly 5.X satisfies (it) and (i2). Now suppose Z g Y g X and Z is an 5.X-subset. Then Z u B e S and so Z u B is contained in a maximal 5-subset B' of S - (X - Y). Now B' = Z' u B where
Z' Q Z and Z' o (S - Y) = 0. Evidently Z' is a maximal 5.X-subset of Y containing Z. Thus 5.X satisfies (IB1).
We now check that 5.X satisfies (IB2). Let Bt and B. be maximal 5.X subsets of a subset Y of X and suppose x e B1 - B2. Then B 1 u B and B2 U B are maximal 5-subsets of S - (X - Y). Since x e (B1 u B) - (B2 u B), there is an element y of (B2 u B) - (B1 u B) such that ((B1 u B) - x) u y is a maximal 5-subset of S - (X - Y). Clearly y e B2 - B1. Moreover, ((B1 u B) - x) u y = ((B 1 - x) u y) u B, and so (B 1 - x) u y is a maximal 5.X-subset of Y. Hence .5.X satisfies (IB2) and the proposition is proved.
In the preceding section it was suggested that one of the problems with attempting to define orthogonality for independence spaces may be that the class of independence spaces is too restricted. In this section we shall confirm
this by showing that (3.3) is an involution on the class of B-matroids. In order to achieve this objective we shall consider certain aspects of the closure-operator approach to infinite matroids as developed by Klee (1971). We defer to the exercises consideration of an equivalent approach that was adopted by Higgs (1969a, b, c). An operator f on a set S is a function from 2s to 2s satisfying
(cl1) X sf(X) for all X 9 S. (c12) X g Y S implies f (X)
f (Y).
We shall concentrate primarily on operators satisfying the following two additional conditions. (c13) f (f (X)) = f (X) for all X g S.
(cl4) IfX, Ys Sandpef(Y)-f(Y-X),then xef((Y-x)up)forsome xinX. Such operators will be called idempotent-exchange- or IE-operators.
It is straightforward to check that (cll){c13) and (cl4') provide another cryptomorphic axiom system for finite matroids to add to the many presented in White (1986).
82
James Oxley
For an operator f on a set S, let f * be defined, for all X c S, by
f*(X)=Xv{x: xOf(S-(Xux))}. 3.2.3. Proposition. If f is an operator on a set S, then f * is an operator on S and (f *)* = f Moreover, if f is an IE-operator, so is f *. The proof of this is left as an exercise. Again, for f an operator on S and X c S, the restriction fx and contraction f' of f to X are defined, for all Y S; X, by fx(Y) = f (Y) n X, and f x(Y) = f(Yu (S - X)) n X respectively.
3.2.4. Proposition. If f is an operator on a set S and X s S, then both fx and f x are operators on X, and (f *)x = (f x )* and (fx)* = (f *)x. Furthermore, if f is an IE-operator, so are both fx and fx.
Proof It is not difficult to check that both fx and f x are operators on X and that if f is an IE-operator on S, then both fx and f x are IE-operators on X. Now, if Y c X, then
(f *)x(Y) = f *(Y) n X =(Yu{yeS: YOf(S-(Yuy))})nX
=Yu{yeX: yOf((S-X)u(X-(Yuy)))} = Yu{yeX: YOfx(X -(Yuy))} =(fx)*(Y). Thus (f *)x = (f x)*. If we replace f by f * in this equation, then, by Proposition 3.2.3 we get fx = ((f *)x)*. Hence, by Proposition 3.2.3 again, (fx)* = (f *)x,
and the proof of the proposition is complete. Let f be an operator on a set S and suppose X c S. Then X is independent if x O f (X - x) for all x in X; otherwise X is dependent. We call X spanning or non-spanning according as f(X) = S or f(X) S; and X is a basis if X is both independent and spanning. A minimal dependent set is a circuit and a maximal non-spanning set is a hyperplane. Clearly when S is finite and f is an IE-operator, f is the closure operator of a (finite) matroid on S and then the definitions above are consistent with the usage of these terms in White (1986). We leave the routine proofs of the next two results as exercises. 3.2.5. Proposition. If f is an IE-operator on S, then the sets of bases, maximal independent sets, and minimal spanning sets are identical.
3.2.6. Proposition. If f is an operator, then (i) the f-independent sets are precisely the complements of the f *-spanning sets;
(ii) the f-circuits are precisely the complements of the f *-hyperplanes; and (iii) the f-bases are precisely the complements of the f *-bases.
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The next proposition shows that B-matroids can be axiomatized in this operator framework. This important fact will be used in the discussion of orthogonality for B-matroids. We require the following lemma. 3.2.7. Lemma. Let f be an IE-operator on a set S and 5 be the collection of f-independent sets. If X c S and B is a maximal.-subset of X, then f (B) = f (X).
Moreover, if I g S and l e f, then f (I) =I u {x: I u x 0 f}.
Proof. As B is a maximal f-subset of X, by Propositions 3.2.4 and 3.2.5, ff(B) = X. Thus X c f (B) and so, by (c12) and (c13), f (X) f (f (B)) = f (B). f (X), hence f (B) = f (X) as required. If I E f and 1 u x 0 5, then y e f ((1 u x) - y) for some element y of I u x. If y = x, then x E f (1). But if y :Ax, we still get x E f (I) by (c14') because y o f (1- y). Therefore, f (1) 2 1 u {x: I u x 0 .f}. Now, if x e f (1) - I, then I u x is not f-independent; that is, I u x J. Hence f (I) c I u {x: I u x O f}, and the required result follows. But, by (cl2), f (B)
3.2.8. Proposition. Suppose that MB(S) is a B-matroid having .f as its collection
of independent sets and let f be defined, for all subsets X of S, by
f (X) = X u {x: I u x o f for some I
X such that I e f}.
(3.6)
Then f is the unique IE-operator on S having 5 as its collection of independent sets, and, since MB(S) is a B-matroid, f satisfies (IB1). Conversely, if f is an IE-operator on S such that the set 5 of f-independent sets satisfies (IB1), then f is the set of independent sets of a B-matroid on S.
Proof. Suppose that MB(S) is a B-matroid and that f is defined as in (3.6). Then clearly f satisfies (clI) and (cl2). We show next that for all X c S, _ Xu{x: XuxOf}, if Xef,
f(X)
f(Ix), if Ix is a maximal f-subset of X.
(3.7)
If X e f and 1 u x 0 f for some I c X, then X u x 0 J. Thus (3.6) and (3.7) agree when X eJ. If X f and Ix is a maximal f-subset of X, then, by (cl2), f (X) 2 f (Ix). We now show that the reverse inclusion also holds. If x e X - IX, then Ix u x 0 f, so x e f (Ix) and hence, f (Ix) 2 X. Next assume that x e f (X) - X. Then I u x f for some f-subset I of X. If 1x u x o f, then x e f (1 x ), so
suppose that I x u x e f. Then Ix u x is a maximal f-subset of X u x. Moreover, I is contained in a maximal f-subset B of X u x, and x 0 B. Thus x e (Ix u x) - B, and so, by (IB2), there is an element y of B - (Ix u x) such that ((IX u x) - x) u y = Ix u y is a maximal f-subset of X u x. But Ix u y s X and so the choice of Ix is contradicted. Thus (3.7) is established. From (3.7), it is easy to see that f satisfies (cl3). To show that f satisfies
(cl4'), suppose X c Y and pef(Y)-X and pOf(Y-X). Let Iy-x be a
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maximal 5-subset of Y - X. Then Il. - X is contained in a maximal 5-subset ly of Y, and, as p e f (Y) = f (11,), we have I r u p 5 and so I,, is a maximal
f-subset of Yup. Now pOf(Y-X)= f(Il,_X), hence Il._Xupe5, and so Il. _ X u p I'1, a maximal f-subset of Yu p. If there is an element x of X such that x 0 I',,, then x e f (Y) s f (Yu p) = f (I'1) = f ((Y - x) u p); that is, (c14') holds. Thus assume that X I ' . Then 1'1 = I y _ X u X u p Il _ X U X Il. This is a contradiction since both Ij, and I'Y are maximal f-subsets of Yu p. We conclude that f satisfies (cl4').
The fact that S is precisely the collection of f-independent sets follows without difficulty from (3.7). Moreover, by Lemma 3.2.7, f is the unique IE-operator on S having 5 as its collection of independent sets.
To prove the converse, let f be an IE-operator on S having 5 as its collection of independent sets and suppose that S satisfies (IB1). Evidently S satisfies (il) and (i2). Therefore to show S is the collection of independent sets of a B-matroid, it remains only to check that (IB2) holds. Note that, by Lemma 3.2.7 and (IB1), f is defined as in (3.7) for all subsets of S. Suppose X S; S and let B, and B2 be maximal 5-subsets of X. Assume that x e B1 - B2
and, for all y in B2 - B 1, the set (B1 - x) u y 0 J. Then, by (3.7), B2 - B 1 s f (B, - x). Hence f (B, - x) = f ((B 1- x) u B2) 2 f (B2) 2 X. Hence
x e f (B1 - x); a contradiction of the fact that B1 is f-independent. This completes the proof of Proposition 3.2.8.
If MB(S) is a B-matroid and f is defined as in (3.6), then we call f the closure operator of MB(S).
3.2.9. Proposition. Let f be the closure operator of a B-matroid MB(S) having S as its collection of independent sets. Then fX and f X are the closure operators of MB(X) and MB.X respectively.
Proof. It is clear that f I X is the collection of fX independent sets, hence by Proposition 3.2.8, fX is the closure operator of MB(X). Now suppose that
YeS.X, but Y is not fX-independent. Then YuBef for some basis B of MB(S - X), and y e f X(Y - y) for some y in Y Thus y e f ((Y - y) u (S - X)) and, as S - X 9 f (B) c f ((Y - y) u B), it follows by (cl2) and (c13) that f ((Y - y) u (S - X)) c f ((Y - y) u B). Therefore y e f ((Y - y) u B). This contradicts the fact that Yu B e J. Thus, if Ye f.X, then Y is f X-independent. On the other hand, if Y is f X-independent and Yo 5.X, then Yu B S where B is a basis of MB(S - X). As B e.S, by (IB1), Yu B contains a maximal f-subset Y u B where Y'$ Y By Lemma 3.2.7, f (Y' u B) = f (Yu B). Thus, if y e Y - Y', then that is, Y is not f x-independent. This contradiction implies that if Y is f X-independent, then YeS.X. We conclude that the set of fX-independent sets equals S.X and
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hence that fX is the closure operator of MB.X. This completes the proof of Proposition 3.2.9.
We are now in a position to prove that the class of B-matroids is closed under the natural orthogonality function (3.3). 3.2.10. Theorem. Suppose that MB(S) is a B-matroid having f as its closure operator. Then f * is the closure operator of a B-matroid on S and its collection 5* of independent sets is given by 5* = {X: S - X contains a basis of MB(S)}. Proof By Proposition 3.2.3, f * is an IE-operator on S. Now, by Propositions 3.2.3 and 3.2.6(i), the f *-independent sets are precisely the complements of the f-spanning sets. In addition, by (3.7), a set is f-spanning if and only if it contains an f-basis. Therefore 5* is as given. It follows from Proposition
3.2.8 that to complete the proof we need to show that 5* satisfies (IB1). Suppose that Yc X 9 S and Yes*. Then S - Y is f-spanning and so X - Y is f '-spanning. But, by Proposition 3.2.9, f' is the closure operator of the B-matroid MB.X and so, by (3.7), X - Y contains an f X-basis B. Thus X - B 2 Y and, by Proposition 3.2.6(iii), X - B is an (f ')*-basis. Therefore, since (f X )* = (f *)X by Proposition 3.2.4, it follows that X - B is a maximal f *-independent subset of X containing Y. Thus 5* satisfies (IB1), and so, by Proposition 3.2.8, Theorem 3.2.10 is proved.
The attention in this chapter has been concentrated on two classes of infinite matroids, the classes of independence spaces and of B-matroids. The motivation for the study of independence spaces was algebraic. The last result of this chapter provides additional motivation for the study of B-matroids, showing that this is a particularly natural class of infinite matroids to examine. Let S be an arbitrary infinite set and suppose that on every subset W of S we have a distinguished class -9W of pre-independence spaces so that (3.1) and (3.2) give well-defined operations of restriction and contraction on -9W such that if Z g W, the restriction or contraction of a member of -9W is in -qZ;
and the function defined by (3.3) is an involution on -9W.
(3.8)
(3.9)
Evidently, if for all W g S, we let 2W be the set of B-matroids on W, then (3.8) and (3.9) hold. In fact, for every choice of -9W subject to these conditions,
the members of 2W are B-matroids. 3.2.11. Theorem. IfS is an infinite set, then the largest class ofpre-independence spaces defined on S and all its subsets such that (3.8) and (3.9) are satisfied is the class of B-matroids.
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For the proof of this result the reader is referred to Oxley (1978a). Note, however, that there is an error in Oxley (1978a) in that only Theorem 3.2.11
above is proved, although a stronger result is stated there. Whether this stronger result is true is an unsolved problem (see Exercise 3.22).
Exercises The more difficult exercises are marked with an asterisk. Section 3.1 3.1.
3.2.
(a) Extend the closure axioms (cllHc14) of Chapter 2 of White (1986) to give a closure-operator definition of an independence space. (b) Show that, by adding (B4) to (blHb3) of White (1986), one gets an axiom system that is cryptomorphic to the system (il), (i2), (i3'), and (14). (c) Show that (b) holds with (b3) replaced by (b3.1) of White (1986). (a) Let I be a set and {Si: i e I} be a collection of pairwise disjoint sets. For all i in I, let Mi(Si) be an independence space having Ji as its collection
of independent sets. Show that I U Ui: Ui e. is the collection of i
I
independent sets of an independence space on U Si. This independence iEI
space is called the direct sum of the independence spaces Mi(Si) (i c 1). *(b)
3.3. 3.4.
(Las Vergnas, 1971; Mason, 1970; Bean, 1976) Let M(S) bean independence
space for which M*(S) is also an independence space. Show that M(S) is the direct sum of a collection of finite matroids. Let M(S) be an independence space and suppose that X cc S. If the restriction of a pre-independence space is defined as in (3.1), show that . r.X = (5* I X)*. (Las Vergnas, 1971) Let M(S) be an independence space and define f+ to be
the collection of subsets X of S such that for all Ycc X there is a basis of M(S) disjoint from Y. (a) (b) (c) *(d)
Show that 5+ is the collection of independent sets of an independence space on S. Prove that =5+ and 5+IX =(S.X)* for all X cc S. Give an example to show that it is not necessarily the case that (5 + )+ _
Let f be the closure operator of M(S); that is, for all X c S,
f(X)=Xu{x: Iux05 for some I gX such that Ie5}. Show that (,f +)+ = 5 if and only if for all finite subsets X of S and all elements x of S - f (X), there is a set Y containing X such that S - Y is finite and x O f (Y). *3.5.
(Piff, 1971) Let X be a family of subsets of a set S and suppose that every
*3.6.
element of S is in finitely many members of T. Use Rado's selection principle to show that the set of partial transversals of 37 is the collection of independent sets of an independence space on S. (Piff, 1971) Prove that if M(S) is an independence space such that for all X cc S
the matroid M(X) is graphic, then there is a graph F such that M(S) is isomorphic to Mr.
Infinite Matroids *3.7.
87
Show that the following are equivalent. (a) The Axiom of Choice. (b) (Zorn's lemma) If every chain in a non-empty partially ordered set possesses (c)
an upper bound, then the set has at least one maximal element. (Tukey's lemma) If ° ' is a non-empty collection of sets satisfying (i2) and (I4), then ' has a maximal member.
Section 3.2 3.8. Higgs's approach to infinite matroids (Higgs, 1969a) uses the idea of a derived
set operator. The series of problems below links this approach to that taken above. For an operator f on a set S, we define Of: 2S-,2S by Of(X) = {x: x e fix - x)}. (a) Show that of satisfies the following conditions.
(D1) If X YES, then Of(X)Of(Y). (D2) If X SS and xeof(X), then xeOf(X -x). (b) Show that for all X c S, f (X) = X u 01(X). (c) Let o be a function on 2S satisfying (DI) and (D2). Define f: 2S-*2S by f (X) = X u o(X). Show that f is an operator on S and that of = o. (d) Characterize IE-operators f in terms of properties of the corresponding derived set operators of. (e) If X c S and f is an operator on S, find of., of , and ofx in terms of of. (f) For an operator f, characterize f-independent sets, f-spanning sets and f-bases in terms of of. 3.9. Axiomatize B-matroids in terms of their collections of bases. 3.10. Show that Proposition 3.2.5 does not hold for arbitrary operators by giving an example of an operator whose sets of bases, maximal independent sets, and minimal spanning sets are all distinct. 3.11. Show that an IE-operator is not uniquely determined by the pair consisting of its collection of independent sets and its collection of spanning sets. 3.12. Consider the following conditions on an operator f on a set S. (c13')
(C)
(H)
If X, Y s S and X g f (Y), then f (X u Y) s f(Y). If Y s S and p e f (Y), then there is a minimal subset U of Y such that p e f (U) and U is independent. If Y c S and p e S - f (Y), then there is a maximal superset V of Y such that p o f (V) and V u p is spanning.
Show that (a) f satisfies (c13') if and only if f satisfies (c13); (b) f satisfies (C) if and only if f * satisfies (H); (c) if f satisfies (cl4') and p, Y, and U are as in (C), then U up is a circuit; (d) if f satisfies (c13) and p, Y, and V are as in (H), then V is a hyperplane.
3.13. Find an example of a B-matroid MB(S) and an operator g on S so that g is not the closure operator of MB(S), yet MB(S) and g have the same collections of independent sets. 3.14. If f is the closure operator of a B-matroid, show that f satisfies (C) and (H).
88
James Oxley
3.15. Let f be an IE-operator on a set S and suppose that f satisfies (C). Show that the collection l of f-circuits satisfies (cI), (c2), and (c3.1) and that, for all X c S,
f(X)=Xu{x: x e C Xux for some C in'}. 3.16. (Higgs, 1969a) Let f be an IE-operator on a set S and suppose that f satisfies both (C) and (H). (a) (Unsolved) Is f the closure operator of a B-matroid on S? (b) (Jnsolved) If X s S, does fX satisfy (H)? (c) If X si S, does fX satisfy (C)? (d) Show that if (a) holds, then so does (b). 3.17. Let f be an IE-operator on a set S. Suppose f satisfies (C) and B is an f-basis.
If x e S - B, use the result of Exercise 3.15 to prove that there is a unique circuit C such that x e C s B u x, and that, if y e B, then (B u x) - y is a basis if and only if y e C. *3.18. (Oxley, 1978a) Use the results of Exercises 3.14, 3.15, and 3.17 to show that if B1 and B2 are bases of a B-matroid and b, a B, - B2, then there is an element
b2 of B2-B1 such that both (B1-b1)ub2 and (B2-b2)ub1 are bases. *3.19. (Higgs, 1969c) Let F be a graph that may be finite or infinite, and let A(r) consist of all finite cycles of F together with all two-way infinite paths in Y. Use the results of Exercises 3.14 and 3.15 to show that .*''(I) is the set of circuits of a B-matroid on the edge set of F if and only if I has no subgraph homeomorphic to the graph in Figure 3.2. Figure 3.2.
*3.20. (Matthews and Oxley, 1977) Let F be a graph that may be finite or infinite, and let .N'(F) be the set of subgraphs of I that are homeomorphic to one of the five graphs shown in Figure 3.3 (where an arrow denotes a one-way infinite path). Use the results of Exercises 3.14 and 3.15 to show that . V(F) is the set of circuits of a B-matroid on the set of edges of r. A detailed discussion of the properties of the finite matroids that arise in this way may be found in Chapter 4 of this volume. *3.21. (Higgs, 1969b) Show that if the Generalized Continuum Hypothesis holds, then all bases of a B-matroid are equicardinal.
Infinite Matroids
89
Figure 3.3.
0 3.22. Let S be an infinite set and -9 be a class of pre-independence spaces defined on S and all its subsets such that (3.8) and (3.9) are satisfied. (a) Show that -9 need not equal the class of B-matroids defined on all subsets of S.
(b) (Unsolved) Does (a) remain true if we insist that .9 contains the class of all those independence spaces that are defined on some subset of S? References Asche, D. S. (1966). Minimal dependent sets, J. Austral. Math. Soc. 6, 259-62. Bean, D. W. T. (1976). A connected finitary co-finitary matroid is finite, in Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory and Computing, Congressus Numerantium 17, pp. 115-19. Utilitas Mathematica, Winnipeg. Bleicher, M. N. & Marczewski, E. (1962). Remarks on dependence relations and closure operators, Colloq. Math. 9, 209-11. Bleicher, M. N. & Preston, G. B. (1961). Abstract linear dependence relations, Publ. Math. Debrecen 8, 55-63. Brualdi, R. A. (1969). Comments on bases in dependence structures, Bull. Austral. Math. Soc. 1, 161-7. Brualdi, R. A. (1970). Admissible mappings between dependence spaces, Proc. London Math. Soc. (3) 21, 296-312. Brualdi, R. A. (1971). On families of finite independence structures, Proc. London Math. Soc. (3) 22, 265-93. Brualdi, R. A. & Scrimger, E. B. (1968). Exchange systems, matchings and transversals, J. Comb. Theory 5, 244-57. d'Ambly, C. G. (1970). Whitneysche Abhangigkeitsstrukturen, in Combinatorial Mathematics and its Applications I, Colloq. Math. Soc. Janos Bolyai 4, pp. 25-37. North-Holland, Amsterdam. Dlab, V. (1965a). Axiomatic treatment of bases in arbitrary sets, Czechoslovak Math. J. 15 (90), 554-64. Dlab, V. (1965b). The role of the `finite character property' in the theory of dependence, Comment. Math. Carolin. 6, 97-104. Glazek, K. (1979). Some old and new problems in independence theory, Colloq. Math. 42, 127-89. Gyori, E. (1975). Vegtelen matroidok osszehasonlitasa (Comparison of infinite matroids), Mat. Lapok 26 (34), 311-17. Gyori, E. (1978). On the structures induced by bipartite graphs and infinite matroids, Discrete Math. 22, 257-61. Higgs, D. A. (1969a). Matroids and duality, Colloq. Math. 20, 215-20. Higgs, D. A. (1969b). Equicardinality of bases in B-matroids, Can. Math. Bull. 12, 861-62. Higgs, D. A. (1969c). Infinite graphs and matroids, in Recent Progress in Combinatorics, Proceedings of the Third Waterloo Conference on Combinatorics, pp. 245-53. Academic Press, New York. Hughes, N. J. S. (1963). Steinitz' exchange theorem for infinite bases, Compositio Math. 15, 113-18.
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Hughes, N. J. S. (1965). Steinitz' exchange theorem for infinite bases II, Compositio Math. 17, 152-5.
Klee, V. (1971). The greedy algorithm for finitary and confinitary matroids, in Combinatorics, Proceedings of Symposia in Pure Mathematics 19, pp 137-52. American Mathematical Society, Providence, Rhode Island. Las Vergnas, M. (1971). Sur la duality en theorie des matroides, in Thiorie des Matroldes, Lecture Notes in Mathematics 211, pp. 67-85. Springer-Verlag, Berlin, Heidelberg, and New York. Las Vergnas, M. (1972). Problemes de couplages et problymes hamiltoniens en theorie des graphes, Thesis L'University Paris VI. Maeda, F. & Maeda, S. (1970). Theory of Symmetric Lattices, Springer-Verlag, Berlin, Heidelberg, and New York. Mason, J. H. (1970). A characterization of transversal independence spaces, in Combinatorial Structures and Their Applications, Proceedings of the Calgary International Conference, pp. 257-9. Gordon & Breach, New York. Matthews, L. R. & Oxley, J. G. (1977). Infinite graphs and bicircular matroids. Discrete Math. 19, 61-5. McDiarmid, C. J. H. (1975). An exchange theorem for independence spaces, Proc. Amer. Math. Soc. 47, 513-14. Mirsky, L. (1971). Transversal Theory. An Account of Some Aspects of Combinatorial Mathematics, Academic Press, London and New York. Oxley, J. G. (1978a). Infinite matroids, Proc. London Math. Soc. (3) 37, 259-72. Oxley, J. G. (1978b). Infinite matroids and duality, in Problemes Combinatoires et Thiorie des Graphes, Colloques Internationaux du C.N.R.S 260, pp. 325-6. Centre National de la Recherche Scientifique, Paris. Perfect, H. (1969). Independence spaces and combinatorial problems, Proc. London Math. Soc. 19, 17-30. Piff, M. J. (1971). Properties of finite character of independence spaces, Mathematika 18, 201-8. Rado, R. (1949). Axiomatic treatment of rank in infinite sets, Can. J. Math. 1, 337-43. Rado, R. (1966). Abstract linear dependence, Colloq. Math. 14, 257-64. Robert, P. (1967). Sur l'axiomatique des systemes gynyrateurs, des rangs etc., C.R. Acad. Sci. Paris Sir. A 265, 649-51. Robertson, A. P. & Weston, J. D. (1959). A general basis theorem, Proc. Edinburgh Math. Soc. (2) 11, 139-41. Simoes-Pereira, J. M. S. (1974). Matroids, graphs and topology, in Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory and Computing, Congressus Numerantium 10, pp. 145-55. Utilitas Mathematica, Winnipeg. Wagstaff, S. S. Jr. (1973). Infinite matroids, Trans. Amer. Math. Soc. 175, 141-53. Welsh, D. J. A. (1976). Matroid Theory, London Mathematical Society Monographs 8, Academic Press, London, New York, and San Francisco. White, N. (ed.) (1986). Theory of Matroids, Cambridge University Press. White, N. (ed.) (1987). Combinatorial Geometries, Cambridge University Press.
4
Matroidal Families of Graphs J. M. S. SIMOES-PEREIRA
4.1. Introduction The connections between graph theory and matroid theory can be traced back to the study of graphic matroids, which were introduced by Whitney (1935) and have been extensively investigated (see Chapters 1, 2, and 6 of White, 1986). We recall that a matroid is graphic if it is isomorphic to the polygon matroid of some graph. In this chapter we present some recent results that give a new setting to the relations between graphs and matroids. In the light of this setting, the polygon matroid appears as the simplest and best known object among an uncountably infinite collection of similar objects. The fundamental concept that we want to introduce is the concept of a `matroidal family of graphs'. The precise definition is given below. According to this definition, the collection of all polygons is a matroidal family of graphs,
the simplest among the non-trivial ones. Another example of a matroidal family of graphs is the collection of all bicycles, where a bicycle is a connected
graph with two independent cycles and no vertex of degree less than two, that is to say, a bicycle is a graph homeomorphic to one of the graphs J00, J0.0, or J0-0 pictured in Figure 4.1. As we shall see, there are uncountably many matroidal families of graphs; the subject is virtually unexplored and this chapter is just a brief introduction to this fascinating new field. Figure 4.1.
><) >0-< Jou
JO.0
Jo-o
92
J. M. S. Simoes-Pereira
4.2. Matroidal Families of Graphs: a Definition Throughout this chapter, IF denotes a graph, V(F) its vertex set, and E(F) its edge set. We shall consider simple graphs, unless otherwise stated. If F' and F are graphs, then F' = F means that they are isomorphic, F' = F means that they are coincident, 17< F means that F' is isomorphic to a subgraph of F,
and F'S: F means that I'' is a subgraph of F. Sometimes the distinction between < and c is convenient. Behzad & Chartrand (1971) is our graph theory reference. A matroidal family of graphs is a non-empty collection P of finite, connected graphs with the following property: given an arbitrary graph F, the edge sets
of the subgraphs of F that are isomorphic to some member of P are the circuits of a matroid on E(F). The matroid defined in this way by the matroidal
family P on the edge set of the graph F will be denoted by P(F'). The assumption of connectedness in the definition is important. If we drop the requirement of connectedness, then, for each positive integer q, the collection P of all graphs (connected or not) with q edges is a matroidal family, P(I') being a uniform matroid. As examples, we present now the four matroidal families of graphs that were discovered first; they will be denoted throughout by Po, P1, P2, and P3. The family PO consists of one graph only, namely the complete graph on two points, K2. The family Pl consists of all polygons or cycles for n > 3. (Polygon and cycle are here essentially synonyms but `cycle' conveys the idea of a polygon that is a proper subgraph of some graph.) The family P2 consists of all bicycles. The family P3 consists of the even polygons of length at least four and the bicycles with no even polygon. In Figure 4.2, IF = K4 and E(17) = {e1, e2, ..., e6 }. The circuits of P0(F) are the sets {e1} for i = 1, 2,..., 6; the circuits of Pl (1') are {e1, e2, e5 }, {e3, e4, e5 }, {e2, e3, e6}, {el, e4, e6}, {el, e2, e3, e4}, {e1, e3, e5, e6}, and {e2, e4, e5, e6};
the circuits of P2(F) are the sets E(F) - {e,} for i = 1, 2, .., 6; the circuits of P3(F) are {el, e2, e3, e4}, {e1, e3, e5, e6}, and {e2, e4, e5, e6}. Among matroidal families of simple graphs, PO is the only one that is finite, as proved in Simoes-Pereira (1973). If IF is simple, P1(F) is the polygon Figure 4.2.
Matroidal Families of Graphs
93
matroid of IF and P2(F) is called the bicircular matroid of IF. If we allow multiple edges in IF, then, in the definition of P1 we should write n >, 2 instead of n >, 3; if we also allow loops, then we should write n >, 1. Similar
considerations apply to P2 and P3. Note that generalizing definitions or results in this chapter to non-simple graphs may not be straightforward. The matroidal families P1 and P2 were first singled out by a theorem proved in Simoes-Pereira (1972). To state it, recall that two graphs are homeomorphic if both can be obtained from a third graph by a (possibly empty) sequence of edge subdivisions. Define a family P of graphs to be closed under homeomorphism if it has the following property: if a graph F" is homeomorphic to a graph F' and F' e P, then I'" E P. 4.2.1. Theorem. The only matroidal families of graphs closed under homeomorphism are P1 and P2. The proof is based on the next three lemmas. For many results on matroidal families, proofs are graph-theoretic. Some lemmas forbid certain subgraphs or configurations in members of P; other lemmas forbid or require certain
graphs as members of P provided that other graphs belong to P or that P satisfies some prescribed conditions. To prove most of the lemmas we take two graphs F' and F" (possibly, F" - F") of a matroidal family P and, by partly superposing them, we form a graph I ; we may write F = F' u F". The graphs F" and F" are now subgraphs of F and their edge sets are circuits of P(F'). For any edge e e E(F') n E(F"), E(F') u E(F7") - {e} contains a circuit of P(F). By conveniently choosing F' and F7", the edges of F' and F" that coincide
in F, and the edge e, we determine new members of P or are led to a contradiction. In the latter case we deduce that F" and F" cannot be in P. 4.2.2. Lemma. If P is a matroidal family distinct from P0, then none of its members has a pendant edge.
Proof. Suppose the contrary. Let J E P have a pendant edge e. Take F'= F" = J and form F from F' and F" so that F'' and F", as subgraphs of F, have e as their only common edge and e is a bridge of F (Figure 4.3). Since E(F') and E(F") are circuits of P(r), the (circuit) definition of a matroid requires the existence of another circuit contained in E(F') u E(F") - {e} = E(F) - {e}. This circuit is either properly contained in one of
the circuits E(F') or E(F"), contrary to the definition, or it is, as a graph, disconnected, with edges in E(F') - {e} and edges in E(F7") - {e}. Since all graphs in P are connected, this is also a contradiction and thus the lemma is proved.
J. M. S. Simoes-Pereira
94
Figure 4.3. --, I"; ----, I"'.
4.23. Lemma. In any matroidal family P distinct from P, and closed under homeomorphism, there is a member that contains, as a proper induced subgraph, a pendant triangle with an edge incident to one of its vertices.
Proof. Take any J e P and let (u, v) e E(J). Let F' and F" be obtained from J by substituting for (u, v) paths (u, w, x, v) and (u, w, y, x, v), respectively. Let IF have I'' and F" as subgraphs with all edges coincident except for the triangle (w, y), (y, x), (x, w) as pictured in Figure 4.4. Let e = (u, w). Since
E(I') and E(r") are circuits of P(t), the (circuit) definition of a matroid
requires the existence of another circuit contained in E(I'') v E(F") - {e} = E(F) - {(u, w)}. This circuit contains the edge (w, x) and at least one
of the edges (w, y) or (y, x) otherwise it is properly contained in another circuit. By Lemma 4.2.2, it contains in fact (w, y) and (y, x). Since, as a graph,
it must be connected, either it is the triangle, and P = P or it also contains the edge (x, v). This completes the proof. Figure 4.4.
J
r
4.2.4. Lemma. If P is a matroidal family distinct from P, and closed under homeomorphism, then the square with a diagonal, the graph formed by two triangles sharing one vertex, and the graph formed by two triangles linked by an edge belong to P.
Proof. We denote these graphs by J00, J0 0, and J0_0, respectively (Figure 4.1). Let J e P contain the induced subgraph referred to in Lemma 4.2.3. Let (v, x) be the edge in J incident with the pendant triangle (w, y), (y, x), (x, w).
Let I'' - F" - J and form IF so that F' and F", as subgraphs of F, have all edges coincident except those in the pendant triangles. Let e = (v, x) (Figure 4.5a).
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95
Figure 4.5.
(a)
(b)
V
X
X
(C)
Reasoning similar to that used in the preceding lemmas shows that Jo.o C- P; a slightly different construction of IF (Figure 4.5b) shows that Joo u P; finally, substituting a path (v, x'), (x', x) for the edge (v, x) in r' (which is possible because P is closed under homeomorphism), forming IF from I'' and F" as indicated in Figure 4.5c, and setting e = (v, x') shows that Jo_o C- P. This completes the proof of the lemma. We can now prove Theorem 4.2.1: As a consequence of Lemma 4.2.4, the bicycles belong to any matroidal family P closed under homeomorphism and
distinct from P1. By Lemma 4.2.2, any other graph of P contains at least one cycle. By the definition of a matroidal family and the (circuit) definition
of a matroid, it cannot contain only one cycle; if it contains two or more, then it contains a bicycle. Hence P = P2 and thus the theorem is proved. Each of the matroidal families Po, P11 P2, and P3 can be partitioned into one or three classes of homeomorphic graphs. This is one of several properties shared only by these four families. 4.2.5. Property. No member of a matroidal family P distinct from Pa1 P11 P21 and P3 has fewer than three independent cycles. In Po, P1, P2, and P3 no member has more than two independent cycles.
4.2.6. Property. No member of a matroidal family P distinct from Po, P1, P21 and P3 has a vertex of degree less than three. In P11 P2, and P3 each member has vertices of degree two.
4.2.7. Property. If P is a matroidal family distinct from Po, P1, P2, and P31 then no two members of P are homeomorphic.
Property 4.2.7 is an immediate consequence of Property 4.2.6. For the proofs of Properties 4.2.5 and 4.2.6, we refer the reader to Simoes-Pereira (1975a).
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A matroid is called bicircular if it is isomorphic to the bicircular matroid of some graph. In view of Theorem 4.2.1, bicircular matroids are, in a sense, an alternative to graphic matroids. They are studied in detail in Matthews
(1977a; 1978a). Matthews proved, among other results, that bicircular matroids are transversal (see Exercise 4.4) and that the dual of a bicircular matroid is a pathic matroid together with a (possibly empty) set of loops. Pathic matroids, as defined in McDiarmid (1974), are in fact an alternative to the cocycle matroid, itself the dual of the polygon matroid. The connectivity
of P2(F) and its relation to a certain type of connectivity of F has been studied in Wagner (1985). The bases of P2 (F) are investigated in Shull et al. (1989).
Unlike P2(F'), the matroid P3(F) has not yet, as far as I know, received a
comprehensive analysis. It emerged on an equal footing with P2(F') in Simoes-Pereira (1975a) and was probably perceived in Duchamp (1972). It appears in the work of Doob (1973) where P2(F) and P3(F) are related with the matroid represented by the eigenspace that corresponds to the eigenvalue
-2 of the adjacency matrix of the line graph of F. P3(F) appears also in Tutte (1976) within the context of chain groups and in Zaslavsky (1982a) within the context of signed graphs. By allowing multiple edges, loops, and even free loops (which are edges with no endpoints), Zaslavsky (1982a) obtains PA(F'), P2 (F'), and P3 (F) as particular instances of his voltage-graphic matroids. P2(F) appears also in Zaslavsky (1987; 1989) and P3(F) in Wagner (1988)
where it is called the factor matroid.
4.3. Countably Many Matroidal Families Andreae (1978) gave the first description of matroidal families distinct from P01 P11 P2, and P3. Theorem 4.3.1 below is a generalization of his result. Although this theorem may be proved by graph-theoretic methods, we prefer to give a proof due to Lorea (1979), which is not essentially graph-theoretic.
4.3.1. Theorem. Let n and r be integers, n >, 0 and - 2n + I < r < 1. Let P,,,, be the set of all graphs IF such that: (i) ni V(h)I + r = IE(h)I and I V(F')I > 2; and (ii) F is minimal with respect to < among graphs with property (i). Then Pa,, is a matroidal family.
Proof. Let F be an arbitrary graph. Define an integer-valued function f on the power set of E(F'), by setting
X=Q
f(X)=0
for
f (X) = nI V'(X)I + r -1
for X00, X
E(F'),
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97
where V'(X) is the set of vertices of F incident to edges in X. Such a function
is non-decreasing and submodular. It therefore defines a matroid, written Pn,,(I'), on E(t), whose independent sets are the sets X c E(F) such that, d Y 9 X, f (Y) % I YI (see Welsh, 1976, Corollary 8.1.1). Given F we now claim that the circuits of P,,,,(F) are the minimal, nonempty
subsets Z of E(I') for which nj V'(Z)I + r = IZI.
(4.1)
To prove this claim it is enough to show that (a) if Z satisfies (4.1), then it is dependent (b) if Z is a circuit, then it satisfies (4.1). To show (a): from (4.1) we get f (Z) < IZI and therefore Z is dependent in P,,,,(F).
To show (b): suppose Z is a circuit. We must have f (Z) < IZI, that is, nj V'(Z)I + r < IZI.
(4.2)
By hypothesis, Z is a circuit; this therefore excludes the possibility that f (Z) > IZI. Now, if IZI = 1, then the inequality (4.2) and f (Z) = nj V'(Z)I + r -1 > 0 imply nj V'(Z)I + r - 1 = 0 = IZI - 1, which proves (4.1). If
IZI > 1, then let Z' c Z, IZ'I = IZI - 1. The set Z' is independent, hence f(Z') > IZ'I and, therefore, nI V'(Z')I + r > IZI.
(4.3)
Inequalities (4.2) and (4.3) yield IZI >, nj V'(Z)I + r > nI V'(Z')I + r > IZI
which proves (4.1) and, therefore, our claim. To achieve the proof of the theorem it is enough to show that the subgraphs of F induced by the edge subsets Z are connected. Suppose the contrary. Let Z1, Z2, ..., Zk, with k > 2, be the connected components of the subgraph of F induced by Z. Since Z is a circuit, Z1, Z2, ..., Zk are independent, therefore we have nI V(Zj)I + r - 1 ,>IZ; I for i = 1, 2, ..., k. It follows that nj V(Z)I + r -1 >, IZI - (k - 1)(r - 1), which contradicts nj V(Z)I + r = IZI, because
r - 1 < 0. This completes the proof. Note that, in proving this theorem, we became aware of the following fact stated here as a proposition for later reference.
4.3.2. Proposition. If F is such that, with n and r as in Theorem 4.3.1, IE(I )I >, niV(I )I + r, I V(I)I >, 2, then there is a connected graph 1 s F such
that IE(r')I = nI V(F')I + r, I V(I'')I > 2.
The matroidal families P0, P1, and P2 are of the form P,,,,. It is an easy exercise to verify that P1 1 = P2, Pl.o = P1 and, by Lemma 4.2.2, P,,,1 _2n = PO for all n > 0.
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98
Figure 4.6.
0
Figure 4.7.
Given P,,,,, it does not seem feasible to draw a picture that visualizes all the graphs in P,,,,. The reader should, however, pause to check that some well known graphs belong to some of these families. For instance, P2,-2 contains all the wheels; it contains also, for example, K3,4 and the graph pictured in Figure 4.6. For n >, 1, contains K2i and contains K21.+ 1. The graph K44 4 and the graph in Figure 4.7 belong to P2001 which contains all connected, 4-regular graphs (Lemma 4.4.3). Finally, no P,,,,. contains K333 (Exercise 4.9).
4.4. Uncountably Many Matroidal Families Schmidt (1979) was able to use certain subsets of the families P,,,, to construct uncountably many matroidal families. He called these subsets partly closed sets. A collection of graphs Q c P,,,, is called partly closed if the following holds: if X, Ye Q, W = X u Y is different from X and Y, and I E(W)l = ni V(W)I + r + 1, then, for every edge e e E(X n Y), there exists Z e Q such that Z< W - {e}. As an example, the set of all even polygons is partly closed. We denote it by T. We have F P1,0. For more examples, see Lemma 4.4.5.
In what follows, the notations are those of the preceding section and all proofs are essentially due to Schmidt (1979).
4.4.1. Theorem. Let Q c
r <, 0, be a partly closed set and define
P.,,+ 1,Q = Q u P', where P' is the set of all graphs of P,,,,,, that do not contain a subgraph isomorphic to a graph in Q. Then 1,Q is a matroidal family.
Proof. It is enough to see that: (1) By the definitions, Pn,r+1,Q contains only minimal graphs with respect
to <; and (2) If X, YeP,,,,+ 1,Q, W = X u Y, and W is different from X and Y, then, for every edge e e E(X n Y) there exists Z e 1,Q such that Z< W - {e}.
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To prove (2) note first that Proposition 4.3.2 and the minimality of X and Y allow us to write IE(W)I = IE(X u Y)I = I E(X )I + I E(Y)I - I E(X n Y)I >, nI V(X)j + r + nI V(Y)I
+r-nIV(XnY)I -r+ 1 =n(IV(X)I+IV(Y)I-IV(XnY)I)+r+1 = nI V(W)I + r + 1.
(4.4)
We distinguish now: CASE 1. X, Ye Q. If, in I E(W)I >, nj V(W)I + r + 1, the equality holds, then (2) follows from the definition of a partly closed collection. Otherwise, take e e E(X n Y). From Proposition 4.3.2, it follows that there exists Z' C- Pf,r+ 1 such that Z'< W- {e}. Now either Z' c- Pn,r+ 1,Q or there exists Z e Q, Z< Z'
hence Z< W- {e}. CASE 2. X, Ye P,,,r+ 1 By definition of a matroidal family, we find Z' C- Pn,r+ 1, Z'< W - {e}. Then we use the argument of Case 1. CASE 3. X e Q, Yo Q. We have IE(W)I > n( V(X)I + IV(Y)I - I V(X n Y)I) + r + 2, which is the situation found in Case 1. This completes the proof of the theorem.
Recalling that the set T of all even polygons is partly closed and contained in P1,O, it is easy to see that P3 = P1,1,T. As a further exercise, the reader is urged to verify that Pn,r+1 = Pn,r+1,o and that Pn,_2n+1 = Pn,-2n+2,u Where U = Pn,-2n+1 To show that there are uncountably many matroidal families of the form Pn,r+ 1,Q we have only to describe a sufficiently large collection of partly closed sets and show that the families they yield are distinct. This can be achieved by the following lemmas.
4.4.2. Lemma. Let the pairs of integers (n, r) and (n', r') be as in Theorem 4.3.1. Then Pn,r 0 if and only if (n, r) (n', r') and at least one of the inequalities 2n + r > 1 or 2n' + r' > 1 holds.
Proof. For the `only if' part we have trivially (n, r)
(n', r'). Moreover 2n + r < 1 implies 2n + r = 1, hence P,,,, = Pn,1 - 2n = P0. Since the same is
true for (n', r'), if neither of the inequalities in the lemma holds, then we would have Pn,r = Pn',r'
For the `if' part, suppose, without loss of generality, that 2n + r > 1, that is, Pn r Po. It follows that Pn,, is infinite. Since (n, r) (n', r'), we get Pn,r P,,.,,., which proves the lemma.
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4.4.3. Lemma. For every n > 1, P,,,0 contains all connected, 2n-regular graphs.
Proof. For any 2n-regular graph IF, JE(F)I = nI V(F)l. Moreover, if F is connected and 2n-regular and F' is a proper subgraph of F, then there exists e e E(F) - E(F') incident with v e V(F'). Hence IE(F')I < njV(I ')I, which implies
the minimality of F, that is to say, no connected, 2n-regular graph contains properly another such graph. This completes the proof of the lemma.
4.4.4. Lemma. For every n > 2, there are infinitely many non-isomorphic, 3-connected, 2n-regular graphs.
Proof. Consider a path of length q >, 2, say (x0, x1), (x1, x2), ...,
(xq_ 1, Xq)
and add to it 2n - 2 pendant edges at each vertex x1, x2, ..., xq_ 1. Let J denote the tree so obtained. The tree J has all vertices of degree 2n except for the set V' of its endvertices. Form a graph J' by adding new edges to J linking endvertices in such a way that the subgraph C induced by V' is a cycle. J' is 3-connected. Let J1, J2, ..., J2i_2 be copies of J' and form with them a graph I in such a way that, for i k, J, n Jk = Ci = C, where Ci Ji is
the replica of C. This graph
IF,
obtained as a join of all Ji for
i = 1, 2, ..., 2n - 2 is 2n-regular and 3-connected. Since distinct values of q obviously yield non-isomorphic graphs F, the lemma is proved. 4.4.5. Lemma. Let Q P,,,0, n > 2, be a set of 3-connected, 2n-regular graphs. Then Q is partly closed.
Proof. With X, Y, and W as in Theorem 4.4.1, we show that the equation JE(W)I = n!V(W)I + 1 cannot be satisfied, which implies the lemma. For this
purpose, suppose that this equation holds. We have then, by (4.4), IE(X n Y)I = IE(X)I + IE(Y)I - IE(W)I = njV(X n Y)1 - 1. With no loss of generality, we now distinguish two cases, both leading to contradictions: CASE I. IV(X n Y)I = I V(X)I = IV(Y)I. Then we get X n Y by deleting just one edge from X, incident with, say, vertices u and v in V(X). As u and v are the only vertices in X n Y of degree less than 2n, they are connected by an edge in Y too. All graphs here are presumed to be simple. It follows that X = Y, that is, X and Y are coincident, a contradiction. CASE 2. 1 V(X n Y)I < I V(X)I. From the 2n-regularity of X, it follows that X n Y has exactly two vertices, say u and v, of degree less than 2n. It follows also that {u, v} separates X because V(X n Y) - {u, v} 0 and V(X) - V(Y) # 0 and no edge of X may join a vertex of X - Y with a vertex of X n Y other than u or v. This contradicts the 3-connectivity of X.
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4.4.6. Lemma. If Q, Q' c P,,,. are partly closed and Q 0 Q', then Pn,r+ 1,Q # Pn,r+ 1,Q''
Proof. The proof of this lemma is trivial. From the preceding lemmas, we immediately obtain the following.
4.4.7. Theorem. There are uncountably many matroidal families of graphs. More precisely, for every n >, 2, there are uncountably many matroidal families of the form Pn,1,Q where Q is an infinite set of 3-connected, 2n-regular graphs.
Partly closed collections of graphs do not look like a natural concept. But they are. To see this, let us give the following generalization which is due to Walter (1982):
For n >, 0, - 2n + 1 < r < 1, a collection Q g Pn,r is s-partly closed if the following holds: if X, Y, e Q, W = X u Y is different from X and Y, and I E(W)I < nj V(W)I + s, then, for every edge e e E(X n Y), there exists Z e Q such
that Z< W- {e}. The proof of the following result can be found in Walter (1982). 4.4.8. Theorem. Let n > 0, - 2n + 1 5 r 5 s, and Q c P,,,r. Let Nn,S be the set of those graphs in Pn,S that do not contain a subgraph isomorphic to a graph in Q. Then Q u Nn,, is a matroidalfamily if and only if Q is s-partly closed and s < 1.
Another result that is worth mentioning is that the existence of uncountably many matroidal families of graphs can also be proved without requiring the graphs to be simple. This generalization of Theorem 4.4.7 is due to Priiss (1984) whose main result is a characterization of matroidal families in terms of what he calls matroidal functions: finitely additive, submodular, non-decreasing functions that map graphs onto non-negative integers and are zero for edgeless graphs.
4.5. Digraphs and Infinite Graphs Some of the results obtained for matroidal families of graphs have analogues
for digraphs. Research in this area was started by Duchamp (1972), and Matthews (1978b) has extensively developed it. Matthews' proofs are analogous
to those used for graphs as exemplified in section 4.2. Loops and multiple edges are however allowed. A matroidal family of digraphs is defined as is a matroidal family of graphs
but with `digraph' instead of `graph' and `weakly connected' instead of `connected'. From a given matroidal family P of graphs, we can obtain a
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matroidal family of digraphs by taking, for each IT in P, all possible orientations of the edges of F. Obviously, only those families not obtained in this way deserve a study in terms of digraphs. In this context, a matroidal family P of digraphs is called proper if there is some digraph F e P and some edge e of I such that the digraph obtained from IF by reversing the orientation of e is not in P. We refer the reader to Matthews (1977c; 1978b) for a proof that there exist no proper matroidal families of digraphs closed under homeomorphism and for a description of all proper matroidal families of digraphs closed under antihomeomorphism. Note that two digraphs I" and IF" are homeomorphic if both can be obtained from a third one by a (possibly empty) sequence of operations consisting of the substitution of a directed path (u, x), (x, v) for a directed edge (u, v), x being a new vertex; they are antihomeomorphic if both can be obtained from a third one by a (possibly empty) sequence of operations consisting of the addition of two new vertices u' and v' and the substitution of an antidirected path (u, u'), (v', u'), (v', v) for the directed edge (u, v).
Matthews (1979) has also investigated a generalization of the definition
of a matroidal family of graphs where connectedness is not required and infinite, non-simple graphs are allowed as members of the family. He concentrated his efforts on families closed under homeomorphism
and used a definition of infinite matroids in terms of circuits that satisfy the (finite) matroid axioms, namely:
(Cl) No circuit contains properly another circuit; and (C2) If C and C" are distinct circuits and e e C r) C", then there is a circuit C such that C c Cu C" - {e}. His main result is a description of all those generalized matroidal families that can be partitioned into a finite number of classes of homeomorphic graphs. He proves that a generalized matroidal family of graphs has only a finite number of homeomorphic classes when all graphs in the family are finite (but not necessarily connected) or are connected (but not necessarily finite).
It is an open question whether there are generalized matroidal families closed under homeomorphism with infinitely many classes of homeomorphic graphs. Matthews (1977c) conjectures that there are none. Theorem 4.4.7 leads me to conjecture that there are infinitely many. Let our disagreement stimulate the reader to try to settle the question.
Exercises 4.1. 4.2.
Show that P0. P2 and P3 are in fact matroidal families of graphs. (Matthews, 1978b)
(a) Show that, allowing non-simple graphs as members, the family consisting of a single graph with j loops at a single vertex is matroidal.
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4.4.
103
(b) Find other finite matroidal families of non-simple graphs. (Matthews, 1977a) Consider I as a road network with a vehicle traveling along the edges of F with U-turns and reversals at vertices forbidden. Let a position of the vehicle be an edge of F together with the orientation of the edge that specifies in which direction the vehicle is moving. Give an intuitive description of the connection of P2(F) in terms of positions. (Matthews, 1977a) For each v e V(I'), let S,, be the set of edges incident with v. Prove that PZ(I') is isomorphic to the transversal matroid M(J) of the family .f = (Se: v e V(I)) )).
4.5.
(Bryant & Perfect, 1977) Define a circuit space as a matroid in which each base
is contained in a circuit. Prove that, for F simple or non-simple, if PZ(F) is a circuit space, then no base has more than two pendant edges, no vertex of F has degree one, and I is connected. 4.6. (Simoes-Pereira, 1978) Let P be a matroidal family distinct from Po, P P2, and P3 and let m 3 3 be the minimum degree of the vertices in any graph of P. Prove that, for some n _> m, the complete bipartite graph K., belongs to P. 4.7. 4.8.
4.9.
(Matthews, 1977c; Andreae, 1978) Prove that, if a wheel belongs to a matroidal family P, then all wheels belong to P. Prove that any matroidal family containing the wheels contains also the graph pictured in Figure 4.7.
Prove that there is no matroidal family of the form
that contains the
complete bipartite graph K333. 4.10. (Matthews, 1978b) Show that the digraphs pictured in Figure 4.8 form a proper Figure 4.8.
co matroidal family of digraphs. 4.11. Let 113 be the collection of graphs whose members are the polygons and the two-way infinite path. Let 114 be the collection of graphs whose members are the bicycles, the two-way infinite path, and the graphs formed by a cycle and a one-way infinite path starting at a vertex of the cycle. (Bean, 1967; Higgs, 1969b; Klee, 1971) Show that 113 and 114 are matroidal families as defined in section 4.5. 4.12. (Schmidt, 1979) Show that there are infinitely many matroids that cannot have the form P(t) for any graph IF and matroidal family P.
Research Exercises 4.13. Characterize the matroids that are isomorphic to some P(F) where F is a graph and P a matroidal family (see Priiss, 1984). 4.14. Prove or disprove the following conjecture: For each pair of integers n, m with n _> m _> 3, there is at least one matroidal family that contains the complete bipartite graph K,,,,.
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References Andreae, T. (1978). Matroidal families of finite connected nonhomeomorphic graphs exist, J. Graph Theory 2, 149-53. Bean, D. W. T. (1967). Infinite exchange systems, Ph.D. Thesis, McMaster University, Hamilton, Ontario. Behzad, M. & Chartrand, G. (1971). Introduction to the Theory of Graphs, Allyn & Bacon, Boston. Bryant, V. & Perfect, H. (1977). Some characterization theorems for circuit spaces associated with graphs, Discrete Math. 18, 109-24. Doob, M. (1973). An interrelation between line graphs, eigenvalues and matroids, J. Comb. Theory Ser. B 15, 40-50. Duchamp, A. (1972). Sur certain matroides definis a partir des graphes, C. R. Acad. Sci. Ser. A, Paris 274, 9-A, 11. Higgs, D. A. (1969a). Infinite graphs and matroids, in W. T. Tutte (ed.) Recent Progress in Combinatorics, pp. 245-53. Academic Press, New York. Higgs, D. A. (1969b). Matroids and duality, Colloq. Math. 20, 215-20. Klee, V. (1971). The greedy algorithm for finitary and cofinitary matroids, in Combinatorics, Amer. Math. Soc. Publication XIX, pp. 137-52. American Mathematical Society, Providence, Rhode Island. Lorea, M. (1979). On matroidal families, Discrete Math. 28, 103-6. Matthews, L. R. (1977a). Bicircular matroids, Quart. J. Math. Oxford (2) 28, 213-28. Matthews, L. R. (1977b). Matroids on edge sets of directed graphs, in Optimization and Operations Research (Bonn, 1977), Lecture Notes in Economics and Mathematical Systems 157, pp. 193-9. Springer-Verlag, Berlin. Matthews, L. R. (1977c). Matroids from graphs and directed graphs, Ph.D. Thesis, Oxford. Matthews, L. R. (1978a). Properties of bicircular matroids, in Probldmes Combinatoires et Theorie des Graphes (Orsay 1976), Editions du C.N.R.S. 260, pp. 289-90. Centre National de la Recherche Scientifique, Paris. Matthews, L. R. (1978b). Matroids from directed graphs, Discrete Math. 24, 47-61. Matthews, L. R. (1979). Infinite subgraphs as matroid circuits, J. Comb. Theory Ser. B 27, 260-73. McDiarmid, C. J. H. (1974). Path-partition structures of graphs and digraphs, Proc. London Math. Soc. (3) 29, 750-68. Press, V. (1984). A characterization of matroidal families of multigraphs, Discrete Math. 52, 101-5. Schmidt, R. (1979). On the existence of uncountably many matroidal families, Discrete Math. 27, 93-7. Shull, R., Orlin, J. B., Shuchat, A. & Gardner, M. L. (1989). The structure of bases in bicircular matroids, Discrete Appl. Math. 23, 267-83. Simoes-Pereira, J. M. S. (1972). On subgraphs as matroid cells, Math. Z. 127, 315-22. Simoes-Pereira, J. M. S. (1973). On matroids on edge sets of graphs with connected subgraphs as circuits, Proc. Amer. Math. Soc. 38, 503-6. Simoes-Pereira, J. M. S. (1974). Matroids, graphs and topology, in Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory and Computing, Congressus Numerantium 10, pp. 145-55. Utilitas Mathematica, Winnipeg. Simoes-Pereira, J. M. S. (1975a). On matroids on edge sets of graphs with connected subgraphs as circuits - II, Discrete Math. 12, 55-78. Simoes-Pereira, J. M. S. (1975b). Subgraphs as circuits and bases of matroids, Discrete Math. 12, 79-88. Simoes-Pereira, J. M. S. (1978). A comment on matroidal families, in Probldmes Combinatoires et Theorie des Graphs (Orsay 1976), Editions du C.R.N.S. 260, pp. 385-7. Centre Nationale de la Recherche Scientifique, Paris. Tutte, W. T. (1976). On chain-groups and the factors of graphs, Research Report CORR 76/45,
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University of Waterloo.
Wagner, D. K. (1985). Connectivity in bicircular matroids, J. Comb. Theory Ser. B 39, 308-24. Wagner, D. K. (1988). Equivalent factor matroids of graphs, Combinatorica 8, 373-7. Walter, M. (1982). Construction of matroidal families by partly closed sets, Discrete Math. 41, 309-15. Welsh, D. J. A. (1976). Matroid Theory, Academic Press, London. White, N. (ed.) (1986). Theory of Matroids, Cambridge University Press. Whitney, H. (1935). On the abstract properties of linear dependence, Amer. J. Math. 57, 509-33. Zaslavsky, T. (1982a). Signed graphs, Discrete Appl. Math. 4, 47-74. Zaslavsky, T. (1982b). Bicircular geometry and the lattice of forests of a graph, Quart. J. Math. Oxford (2) 33, 493-511. Zaslavsky, T. (1987). The biased graphs whose matroids are binary, J. Comb. Theory Ser. B 42, 337-47.
Zaslavsky, T. (1989). Biased graphs. I. Bias, balance and gains, J. Comb. Theory Ser. B 47, 32-52.
5
Algebraic Aspects of Partition Lattices IVAN RIVAL and MIRIAM STANFORD
5.1. Introduction Two results of the last decade highlight the algebraic advances in the theory of partition lattices. The first answers affirmatively a celebrated conjecture about the place of (finite) partition lattices in lattice theory itself. The second is a compactness result according to which every partition on a finite set can be expressed algebraically in terms of a surprisingly small subset of all of the partitions on that set. Briefly the results are these. In a now famous footnote to his paper `Lattices, equivalence relations, and subgroups', Whitman (1946) conjectured that every finite lattice is lattice embeddable in a finite partition lattice. The conjecture resisted proof for many years. In the late 1970s a positive
solution was announced by Pudlak and Tuma (1977) and in 1980, they published their proof of it. The proof is long and complicated. It follows from a result of Wille (1976) on simple lattices that only a few simple lattices have at most three generators: the 1-element chain 1, the 2-element chain 2, and the 5-element modular, non-distributive lattice M3. Figure 5.1. 1, the 1-element chain; 2, the 2-element chain; M3, the 5-element modular
non-distributive lattice; H4, the lattice of partitions on a 4-element set; PG(2), the lattice of subspaces of a field of characteristic 2.
0
0 1
2
PG(2)
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It is well known too that the lattice of subspaces of a finite projective geometry
is 4-generated. In the early 1970s Wille put forth a bold conjecture that every finite partition lattice is four-generated. This was proved by Strietz (1975). The proof is not particularly difficult once a 4-generating set is discovered. Our aim in this chapter is to survey the background to these two results, as well as related results on the algebraic side of the theory of partitions.
5.2. The Lattice of All Partitions Let S be a non-empty set. A partition on S is a family of non-empty, disjoint subsets of S whose union is all of S. These subsets are called blocks. Of course, partitions correspond to `equivalence relations', a term preferred by some. (For a variety of simple examples, see Chapter 1, Examples and basic concepts, in Volume I of this series (Crapo, 1986).) Let n(s) stand for the set of all partitions on S. Let 0, 0 e II(S). For a, b e S, we write a 0 b if a and b are in the same block of 0. Write 0 i 0 if, for each
a, b e S, a 0 b implies a ' b. Then (II(S), <) (or simply II(S)) becomes an ordered set; in fact, n(s) is a lattice with operations meet (infimum) A and join (supremum) v defined by a(0 A cli)b if and only if a O b and a 0 b
and a(0 v cli)b if and only if there is a positive integer n, a sequence a = ao, a1, a2, ..., a = b of elements of S, and a sequence 01, 02, ..., 0,, of partitions,
each equal to either 0 or 0, such that ao01a1, a102a2, ..., These operations can be extended to any subset of II(S). Thus, for a family (Oi I i e I) of partitions,
a1 A Oi I b if and only if aOi b for each i e 1 ieI
///
and,
a V I b if and only if there is a positive integer n, a sequence a = ao, iel
a1, ..., a = b of elements of S, and a sequence Oi,, Oi2, ..., Oi,,, each a member of the family (0iI ieI), such that ao0i,a1, a10i2a2, ..., a complete lattice. (For an introductory account of lattices, see Chapter 3, Lattices, in Volume I of this series (Faigle, 1986), and, for more elaborate and extensive accounts refer to Birkhoff (1967), Crawley & Dilworth (1973), or Gratzer (1978).) II(S) has a bottom, denoted 0, and a top, denoted 1, prescribed by a O b if and only if a = b and, alb for all a, b e S.
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Besides the self-evident symmetry and duality built into the very definition of lattices, they have a further, rather far-reaching quality: a split personality.
The salient points are these. First, there is a natural order-theoretic aspect to a lattice. Indeed, the relation S defined by
O' if and only ifOv Ii=0 is an order on H(S), that is, a reflexive, antisymmetric, and transitive relation.
Thus, a lattice may be considered as an ordered set in which every pair of elements has a supremum and an infimum. This is the order-theoretic aspect and it has natural consequences for the choice of the mappings and subobjects
that are appropriate for its theoretical development. Second, there is the algebraic aspect which emphasizes the definition of a lattice as an algebra, that is, as a set n(S) with two binary operations v and n defined by these postulates:
eve=6=9n 0;
0vO=ifiv0,0AO=,n9; Bv(0VT)=(OVO)vi, 0A(0 nr)=(9n0)AT; 0v(0 A8)=8=8A(0 v8). This definition, based entirely on the formal properties of the two binary operations v and A, is the algebraic aspect. The two aspects are formally equivalent yet each leads, according to its respective tradition, to different questions, techniques, and theories (cf. Rival, 1978). Our needs in this chapter are mostly served by following the algebraic viewpoint. Thus, the appropriate
morphisms here are mappings preserving both operations v and A. (Such mappings are, a fortiori, order-preserving.) Thus, a lattice embedding of a lattice K into a lattice L is a one-to-one map f of K to L such that, for each a, b e K, f (a n b) = f (a) A f (b) and fl a v b) = f (a) v f (b). An order embedding
of K into L is a one-to-one, order-preserving map of K to L whose inverse, too, is order-preserving (cf. Figure 5.4). Every finite lattice K can be order embedded into the lattice 2" of all subsets of an n-element, where n = IKI. To find familiar classes of finite lattices into which every finite lattice can be lattice embedded is a much, much harder problem! Our primary interest here will be in the algebraic aspect, yet we should certainly exploit the convenience of the order-theoretic aspect for its facility in combinatorial themes.
For ease in working we denote the partition ({a, b}, {c, d}), say, of {a, b, c, d j by (a, b)(c, d), or simply (ab)(cd), and the partition ({a, b}, {c}, {d j)
simply by its non-trivial block (ab). Thus, for instance (cf. Figure 5.2),
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Figure 5.2. II({a, b, c, d}), the lattice of partitions on the 4-element set {a, b, c, d}. I = (abcd)
0 = (ab) A (ac) = (ab) A (cd) _ (ab)(cd) A (ad)(bc) = etc., (abc) = (ab) v (bc) = (ab) v (ac) = etc., (ac)(bd) = (ac) v (bd),
I = (abc) v (cd) = (abd) v (bcd) = etc.
Say that 0 covers in II(S), and write 0 >- 0, if 0 > i > i/i in II(S) implies t = i/i. Each partition 0 )- 0 is called an atom of II(S). Every partition is a join of atoms (see Exercise 5.2). A partition 0 is compact if 0 < V 0i implies ieI
that 0
V 0i for a finite subset F of I. In fact, every atom is compact (see ieF
Exercise 5.3). Thus II(S) is compactly generated, that is, every element of II(S) is the join of compact elements. II(S) is semimodular, that is, for every 0, e II(S), 0 non-comparable to 0, if 0>- 0 A cli then 0 v 0 - 0 (see Exercise 5.5).
In summary, the partition lattice H(S) is a geometric lattice, that is, a complete, semimodular lattice in which every element is the join of atoms. Moreover, n(S) is complemented, that is, for each 0 e n(S) there is j e n(S) satisfying 0 A 0 = 0 and 0 v 0 = 1. 0 is a complement of 0 (see Exercise 5.6). This is an important result in the algebraic theory of partition lattices (Ore, 1942, cf. Gratzer, 1978).
5.2.1. Proposition. H(S) is simple.
A partition lattice II(S) is simple if it has no non-trivial congruence relations besides the identity relation co, defined by 0 - 0(w) if and only if 0 and the all relation i defined by 0 - fi(r) for all 0, 0 E II(S).
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Every element in II(S) is the join of atoms; thus the atoms form a generating set. There would seem little reason to suspect that there are generating sets
for finite partition lattices significantly smaller in size than the number of atoms. It is all the more surprising, therefore, that a set of four partitions can always be found that, by taking joins and meets, generates every other partition. 5.2.2. Theorem. (Strietz, 1975; 1977) Every finite partition lattice is 4-generated.
For instance, such a generating set for H({a, b, c, d)}
is {(abc),
(ab)(cd), (ad)(bc), (acd)}, for the atoms of II({a, b, c, d}) can be expressed as lattice polynomials in terms of these generators (cf. Figure 5.2): (ab) = (abc) A (ab)(cd) (ac) = (abc) A (acd)
(ad) = (acd) A (ad)(bc) (bc) = (abc) A (ad)(bc)
(cd) = (acd) A (ab)(cd)
(bd) = (abd) A (bcd) = ((ab) v (ad)) A ((bc) v (cd)) = (((abc) A (ab)(cd)) v ((acd) A (ad)(bc)))
A (((abc) A (ad)(bc)) v ((acd) A (ab)(cd))).
Here are the general ideas. For convenience we write n,, for H(S), where S = { 1, 2, ..., n}. As every element of 11 is the join of atoms a subset of n,, is a generating set if it generates the atoms of n.. As above, let (i, j) stand
for the atom of Hn whose non-trivial block consists of i and j. In fact, a subset of n,, is a generating set if it generates all atoms (1, i) and (2, j), where 2 < i 5 n and 3 <j < n, for these in turn, generate all atoms, since
(i,j)=((1, i) v (1,j)) A ((2, i) v (2,j))
as long as i and j are both distinct from 1 and 2. For n > 16 satisfying n - 1 (mod 3) this is a generating system for 01 =(1, 2), 02 = (1, 2, 11), (3, 4, 12), (5, 6, 8), (7, 9, 10), ..., (k + 3, k + 4, k + 5),
03 = (1, 4, 6), (3, 5, 9), (2, 7, 12), ..., (k - 2, k, k + 5), ..., (n - 5, n - 3), (n - 2, n), 04 = (2, 3), (4, 8), (1, 5, 10), (6, 7, 13), ..., (k -1, k + 1, k + 6), ..., (n - 4, n - 2), (n
where k = 1 (mod 3) and 10 < k < n - 6. Let T = 11, 2, ..., m} S satisfy 10 < m < n. Then the generators 01, 02, 03, 04 restricted to T determine generators 01, 02, 03, 04 of II.. Strietz (1977) exhibits 4-element generating sets for 114, 115, 116, 117, 118, and 119 separately, each isomorphic to 1 + 1 + 1 + 1 (see Figure 5.3).
Algebraic Aspects of Partition Lattices
111
Figure 5.3.
0 0
0
0
0
0
1+1+1+1
0
0
0
1+1+2
0 0
0
0
1+2+2
0
0
0
1+1+3
Strietz (1975) has also considered the structure of the generating sets as ordered sets. Every partition lattice n > 4, contains a generating set isomorphic to I + 1 + 1 + 1 (see Figure 5.3). 114 contains no generating set isomorphic to 1 + 1 + 2 although n,, does for each n >, 10. Moreover, every H,,, n > 4, has a generating set isomorphic to 1 + 2 + 2, and every IIn, n >, 5, has a generating set isomorphic to 1 + 1 + 3. In fact, Strietz (1977) shows the following.
5.2.3. Theorem. If the ordered set P is a generating set of H,,, n >, 10, then P contains an isomorphic copy of 1 + 1 + 1 + 1 or I + 1 + 2. The generating set {B1, 92, 03, B4 } described above is isomorphic to 1 + 1 + 2,
where 01 < 02. For n > 10, on replacing 01 with
91=(1,2,3) and observing that 0, v 02 = 01 it follows that 011 , 02, 03, 04} is also a generating set of II,, isomorphic to 1 + 1 + 1 + 1. Here is another basic result. It concerns lattice identities, that is, identities of the form x A (y v z) = (x A y) V (X A z)
x A (y v (x A z)) = (x A y) v (x A z)
x=y
(distributivity) (modularity) (triviality).
5.2.4. Proposition. (Sachs, 1961) Any lattice identity that holds in every finite partition lattice must hold in every lattice.
In other words, there are no non-trivial lattice identities that are satisfied by all finite partition lattices. A proof of this result can be pieced together from these steps. Let S be a set (not necessarily finite) and let 11f(S) stand for the sublattice of II(S) consisting of all finite joins of atoms of II(S). A directed down set (alias ideal) in II(S) is a subset D such that x < y and y a D implies x a D and x, y a D implies x v y a D. The set of all directed down sets ordered by set inclusion is a lattice too. If there is a lattice identity that holds in every finite
Ivan Rival and Miriam Stanford
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partition lattice then it must hold in every partition lattice as well (see Exercises 5.9-5.11). The concluding step uses a fundamental result on which we shall dwell longer in the next section. For now we simply state and use it: every lattice is isomorphic to a sublattice of a partition lattice (Whitman, 1946). The proof is complete once we observe that an identity that holds in a lattice also holds in every sublattice (see Exercise 5.12).
5.3. Lattice Embeddings We say that K is lattice embeddable in L if there is a lattice embedding of the lattice K into the lattice L. For convenience, we also identify a lattice embedding f of K into L with the sublattice f (K) of L. But notice that, for a one-to-one order-preserving map f of K to L, f(K) may be a lattice (with respect to the induced ordering of L), in which case we call it an order sublattice. f need not, of course, be an embedding (see Figure 5.4). (This is the order-theoretic versus the algebraic aspect.) The first important result on lattice embeddings of partition lattices is as follows (Birkhoff, 1935). Figure 5.4. (a) An order embedding that is not a lattice embedding; (b) a lattice embedding. f(a v b)
(a)
(b)
5.3.1. Theorem. Every partition lattice is lattice embeddable in the lattice of subgroups of some group.
The most fundamental and far-reaching result on lattice embeddings of partition lattices is due to Whitman (1946). 5.3.2. Theorem. Every lattice is lattice embeddable in a partition lattice.
It was actually conjectured by Birkhoff much earlier (Birkhoff, 1935). The attractive consequence is this.
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113
5.3.3. Corollary. Every lattice is lattice embeddable in the lattice of subgroups of a group.
This corollary was, at one time, advertized (not entirely unconvincingly), in order to market lattices, along side groups, rings, and other mathematical produce. Up to the mid 1970s the preeminent unsolved problem in lattice
theory itself was an apparently technical point that seemed to arise from Whitman's efforts to polish and improve his lattice embedding theorem. Indeed, it is in a footnote to his paper (Whitman, 1946) that this conjecture seems to be first recorded: every finite lattice is lattice embeddable in a finite partition lattice.
Over the next three decades efforts to settle the conjecture were focussed on the two following approaches.
(1) Construct a class of finite lattices (as much like finite partition lattices as possible) into which every finite lattice can be lattice embedded. (2) Construct classes of finite lattices (as many as possible) that can be lattice embedded into finite partition lattices. In spirit, at least, the first approach aimed to settle the conjecture positively by converging on finite partition lattices and the second aimed to settle it by the inevitable march to a counterexample. The first was destined to play the major role. Nevertheless, the second has shed light on the `effectiveness' of lattice embeddings into finite partition lattices. Every element of a (finite) partition lattice is the join of atoms. Every (finite) lattice is order embeddable in a (finite) lattice, say 2", in which every element is the join of atoms (cf. Exercise 5.1). The first major result is recorded in Finkbeiner (1960) who credits it as an unpublished result of Dilworth.
5.3.4. Theorem. Every finite lattice is lattice embeddable in a finite, semimodular lattice.
Actually Finkbeiner showed considerably more. An integer-valued function r defined on a finite lattice L is a (semimodular) rank function if r(OL) = 0,
where OL is the bottom of L, and, for each a, b e L, r(a) < r(b) whenever a < b in L, and
r(a v b) + r(a A b) < r(a) + r(b).
Actually, every lattice has a rank function. The principal result in Finkbeiner (1960) is this:
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5.3.5. Theorem. Every finite lattice L with rank function r can be lattice embedded in a finite, semimodulator lattice K such that r = h I L, where h I L is the restriction of the height function It for K to L.
These semimodular lattices need not be geometric. Although it was not published until much later (Crawley & Dilworth, 1973), Dilworth proved (apparently around 1950) the following:
5.3.6. Theorem. Every finite lattice can be lattice embedded in a finite geometric lattice.
The proof entails and presumes several basic facts about closure operators on a set. A closure operator on a set S is a map of the set of all subsets of S to itself, which assigns to each A S a subset A` 9 S satisfying A - A`, A" = A, and A s B implies A` c B`.
Call a subset A of S closed if A` = A. A closure operator on S satisfies the exchange property if, for each A s S, and for each a, b e S,
ae(Au{b})` and a0A` imply bE(AV{a})`. Recently Gratzer & Kiss (1986) have succeeded in blending the results of Dilworth and of Finkbeiner:
5.3.7. Theorem. Every finite lattice L with rank function r can be lattice embedded in a finite, geometric lattice K such that r = h I L, where h is the height function for K.
In a series of papers Hartmanis (1956, 1959, 1961) sought to exploit a geometric analogy. A partition on a set S is, after all, a family of subsets such that every element of S belongs to exactly one of these subsets and every subset contains at least one element of S. A geometry on S is, by analogy,
a family of subsets ('lines') of S such that every pair of distinct elements ('points') of S is contained in a unique subset and each subset contains at least two elements of S. The analogy may be pursued. A partition of type n (n > 1) on a set S is a family of subsets such that any n distinct elements of S are contained in a unique set and each subset contains at least n elements of S. Let stand for the set of all partitions of type n on S. is a subset of II(S) from which it inherits an order. Note that LP1(S) = II(S) and LP2 (S) is a geometry on S (cf. Tuma, 1980). The principal result is this (Hartmanis, 1961):
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5.3.8. Theorem. For each n > 2, and for every finite lattice L, there is a finite set S such that L can be lattice embedded in LPn(S). In a now famous paper, 'Algebras whose congruence lattices are distributive', seminal to the modern theory of universal algebra, Jonsson (1967) proved, inter alia, the following.
5.3.9. Theorem. Every (finite) lattice
is
lattice embeddable in a (finite)
subdirectly irreducible lattice.
A lattice is subdirectly irreducible if it has a unique smallest congruence relation larger than the identity relation co. Every simple lattice is, a fortiori, subdirectly irreducible. This, and another basic property of finite partition lattices, is the motivation for yet another result of this type. Poguntke & Rival (1976) have proved the following.
5.3.10. Theorem. Every finite lattice can be embedded in a finite, 4-generated, simple lattice.
Proof. Let L be a finite lattice and let T(L) stand for the set of all triples (xi, yi, zi) of elements of L such that zi>- yi>- xi. Let L'= L u U {ai, bi }, i
where {at, bt }. {a2, b2 }, ... is a sequence of disjoint pairs of elements (disjoint
also from L), and with order induced by L and the covering relations zi>- ai}- xi and zi> - bi>- xi. Then L' is a finite, simple lattice containing L as a sublattice. The next step in the proof (see Exercises 5.18, 5.19) relies on the following classical result. 5.3.11. Proposition. (Sorkin, 1954; Dean, 1956) Every finite lattice is lattice embeddable in a finite, 3-generated lattice.
The last step in the proof is to construct a lattice L"' = L" u {c, d},
where c, d are distinct elements, disjoint from L", with order induced by L" and 1 >- c } b and c >- d >- 0 (see Exercises 5.20, 5.21).
Of course, the culmination of this approach is the theorem of Pudlak & Tuma (1977, 1980).
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Ivan Rival and Miriam Stanford
5.3.12. Every finite lattice can be lattice embedded in a finite partition lattice.
Proof. The complicated proof relies basically on graph-theoretical and combinatorial techniques. Here is the opening gambit which, interestingly, turns on an order-theoretical aspect of lattices. For a lattice L and elements a 5 b in L, let
La,b={xeL I a
xorb<x}.
La,b is an order sublattice of L (see Exercise 5.22). Define a map fa,b of L to L. b by
if a x x v b if a< x. x
f°'b(x)
fa,b is a join homomorphism of L onto La,b (see Exercises 5.23-5.25). The rest of the proof relies on techniques developed by Pudlak and Tuma to establish
that, whenever L is a finite lattice, lattice embeddable in a finite partition lattice, and a < b in L, then La,b is also lattice embeddable in a finite partition lattice. The starting point, at least, is that every lattice 2" is lattice embeddable in a finite partition lattice (see Exercise 5.26). Their result can also be cast in the language of groups thus.
5.3.13. Theorem. Every finite lattice is isomorphic to a sublattice of the subgroup lattice of a finite group.
The theorem of Pudlak and Tuma settles the long standing conjecture of Whitman which was, after all, the best known and the most sought after in lattice theory. Nevertheless, it is likely that the earlier results, while apparently
concluding less, will survive too for they often invoke important and interesting techniques and provide `effective' lattice embeddings in lattices that are structurally like partition lattices. The Pudlak-Tuma theorem guarantees that for every finite lattice L there is a finite set S such that L can be lattice embedded in H(S). How big is S? In their proof this set S is large. There is, however, a body of results, some of early vintage, others fairly recent, that address this natural question. Hales (1970) concludes that every finite sublattice F of a free lattice can be lattice embedded in a finite partition lattice by extending meet-homomorphisms
of F, element by element, into successively larger partition lattices. The argument proceeds by induction on the length of elements of F and applies an `extension lemma' of Jbnsson (1953). This early paper of Jbnsson, in turn, establishes several striking properties about the lattice embeddings of lattices into partition lattices (albeit infinite ones). Notice that, for 0, 0 e II(S), since
0A0=0n0,
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117
the meet 0 A t/i is easy to compute. In contrast, the join 0 v 0 in n(S) is not so easy to express, for 0 v , is the union of the sequence
0, 0°0, 0°0°0, 0°0°0°0, ... where the `composition' 0-0 is the partition (or equivalence relation) on S defined by a 0 ° >li b if there is c satisfying a 0 c and c >V b. Say a lattice embedding
f of a lattice L into n(S) has type 1 if
f(x) v f(y) =f(x)°f(y),
type 2 if
f(x)vfly) =f(x)-f(y)°f(x),
type 3 if f (x) v fly) = f (x) -fly) °f (x) °f (y), for each x, y e L. In fact (cf. Exercise 5.27):
5.3.14. Theorem. A lattice has a type 2 lattice embedding in a partition lattice if and only if it is modular. The lattice of subspaces of a projective plane has a lattice embedding of type 1 if and only if the plane is Desarguesian.
Finally, we have 5.3.15. Theorem. Every lattice has a lattice embedding of type 3. These results of 1953 did not, of course, necessarily bear on lattice embeddings
to finite partition lattices. For a finite lattice L, let n(L) denote the least positive integer such that there is a lattice embedding of L to a partition lattice II(S), where BSI = n(L). Call a sublattice L of H(S) a {O, 1 }-sublattice if it contains the bottom 0 and
the top 1 of L. It follows from Exercise 5.28 that, if L is a sublattice of L1 x L2 x ... x Lm, then n(L) 5 n(L 1) + n(L2) +
... + n(Lm) - m + 1.
Let L(i, j) be the lattice constructed from chains of size i + 2 and j + 2 by identifying only their bottom and top elements (see Figure 5.5). Ehrenfeucht, Faber, Fajtlowicz & Mycielski (1973) prove
n(L(i, j)) = i+j-1 + r2. J(i+j-2)1. For the modular, non-distributive lattice M with n atoms, they show that p,
where p is the smallest prime satisfying p > n. Building on Dilworth's embedding theorem for geometric lattices, Peele (1982) has shown that certain geometric lattices are lattice embeddable in finite partition lattices. Peele defines an index called the nullity of a geometric lattice L as the number of atoms of L minus the rank of the top of L. (The
Ivan Rival and Miriam Stanford
118
Figure 5.5.
i
Z////\
2
j
2
M
L(i, j)
rank is the least number of atoms whose join is the top. The nullity of 2" is zero and the nullity of M3 is one.) Let L be any lattice obtained from 2" by succesively removing elements covered by the top element while retaining the original order. Every such lattice L with m atoms and of nullity one is lattice embeddable in a partition lattice II(S) satisfying ISO = 2m3 - 5m2 + 4m. (This estimate is, apparently, not best-possible, e.g. m = 4.)
5.4. Postscript
The longstanding lattice embedding problem for partition lattices has been solved. Attention has since focussed on lattices of congruence relations of universal algebras. Gratzer & Schmidt (1963) characterized these lattices (cf. Pudlak, 1976). The result was in some sense analogous to Whitman's original lattice embedding theorem, for it used (possibly infinite) algebras to characterize
even finite (congruence) lattices. The main open problem now is this. Is every finite lattice isomorphic to the congruence lattice of a finite algebra?
Among lattices M, for example, it is known that for n equal to a prime power plus one, n = 7 (Feit, 1983) or n = 11 (Palfy), M" is isomorphic to the congruence lattice of a finite algebra. M13 is the smallest undecided case at this time. There is an attractive equivalent formulation due to Palfy & Pudlak (1980): every finite lattice is isomorphic to the congruence lattice of a finite algebra if and only if every finite lattice is isomorphic to an interval in the
Algebraic Aspects of Partition Lattices
119
subgroup lattice of a finite group.
Exercises Show that the following hold. 5.1. 5.2.
For every finite lattice K there is an order embedding of K into 2", where n = IKI. Every partition in II(S) is the join of a family of partitions each of which contains
precisely one non-trivial block (that is, a block with more than one element of S), and this non-trivial block itself consists of precisely two elements of S. 5.3.
Let 0, 011 02, ... be elements of II(S). If 0 is an atom and 0 5 V 0; then there iEI
is a finite subset F of I such that 0 5 V Oi. ieF
5.4. 5.5. 5.6. 5.7.
Let U(fi) _ {0 e H(S) 10 5 0 < 11 and let T be the set whose elements are the blocks of Then, as lattices, U(ili) =11(T). Let 0, 0 a II(S) satisfy 0 >- 0 and 0 z 0 (that is, 0 A 0 = 0). Then 0 v 0 >- k/i. For each 0 e II(S) there is 0 e II(S) satisfying 0 A 0= 0 and 0 v 0 =1. II(S) is relatively complemented, that is, for each 0 5 T in H(S) there is T 'C- II(S) such that T A T' = 0 and T V T' = 0.
Every 4-element generating set of II({a, b, c, d}) is an antichain. 5.9. II(S) is isomorphic to the lattice of all directed down sets of 111(S). 5.10. Every lattice identity that holds in a lattice also holds in the lattice of all of its directed down sets. 5.11. If there is a lattice identity that holds in every finite partition lattice then this lattice identity must hold in every H1(S) too. 5.12. An identity that holds in a lattice also holds in every sublattice. 5.13. The height function 5.8.
h(a) = max {ICI I C is a chain in L and a is the top of C},
for a semimodular lattice L, is a rank function. 5.14. The set of all closed subsets of a set S, with respect to a closure operator on S, and ordered by set inclusion, is a complete lattice. 5.15. The lattice of all closed subsets of a set S with respect to a closure operator on S satisfying the exchange axiom is geometric. 5.16. The set LP"(S) of all partitions of type n on S is a complete lattice.
Exercises 5.17-5.21 refer to Theorem 5.3.10 et seq. 5.17. L' is a finite, simple lattice containing L as a sublattice. 5.18. Every finite lattice is lattice embeddable in a finite, 3-generated, subdirectly irreducible lattice L". 5.19. In a finite, subdirectly irreducible lattice L" with top 1 and bottom 0, there are elements 1 > a >- b > 0 such that the smallest congruence relation on L" that identifies a and b is the smallest non-trivial congruence relation on L". 5.20. L"' is 4-generated and contains L" as a sublattice.
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5.21. The smallest non-trivial congruence relation on L"' that identifies a and b also identifies 0 and I in L"'. Therefore, L"' is a simple lattice.
Exercises 5.22-5.26 refer to Theorem 5.3.12 et seq. 5.22. La,b is an order sublattice of L. 5.23. fa,b
is an onto map and a join homomorphism, that is, for each x, y e L,
fa,b(x V Y)=fa,b(x) vfa.b(Y)
5.24. Every join homomorphism f of a lattice K to a lattice L such that f (a) = f (b), for some a < b in K, is the composition of the join homomorphism fa,b of K to Ka,b and a join homomorphism g of Ka,b to L. 5.25. Let 2' be a class of lattices containing the lattices 2" of all subsets of an n-element set ordered by inclusion (n >, 1) and containing, for every Lc- 2' and every a < b in L, the lattices La,b. Then 2' is the class of all finite lattices. 5.26. Every lattice 2" is lattice embeddable in a finite partition lattice. 5.27. If f is a lattice embedding of type 2 then L is modular. (Show that, for a, b, c e L with a < c, fl (a v b) A c) s f(a v (b A c)).) 5.28. Let K and L be {0, 1 }-sublattices of n(S) and 11(T) respectively. Then the set of all partitions of 11(S u T) of the form 0 e K, 0 e L, or 0 ° 0 is a {0, 11 -sublattice
isomorphic to the direct product K x L. (Each partition 0 e II(S) can also be considered as a partition of II(S u T) by adjoining to its blocks a singleton block for each element of T - S.)
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